2017

• Paper 1, Section I, F

  #Analysis I

  2017

  Given an increasing sequence of non-negative real numbers $\left(a_{n}\right)_{n=1}^{\infty}$, let

  $$s_{n}=\frac{1}{n} \sum_{k=1}^{n} a_{k}$$

  Prove that if $s_{n} \rightarrow x$ as $n \rightarrow \infty$ for some $x \in \mathbb{R}$ then also $a_{n} \rightarrow x$ as $n \rightarrow \infty$

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• Paper 1, Section II, F

  #Analysis I

  2017

  (a) Let $\left(x_{n}\right)_{n=1}^{\infty}$ be a non-negative and decreasing sequence of real numbers. Prove that $\sum_{n=1}^{\infty} x_{n}$ converges if and only if $\sum_{k=0}^{\infty} 2^{k} x_{2^{k}}$ converges.

  (b) For $s \in \mathbb{R}$, prove that $\sum_{n=1}^{\infty} n^{-s}$ converges if and only if $s>1$.

  (c) For any $k \in \mathbb{N}$, prove that

  $$\lim _{n \rightarrow \infty} 2^{-n} n^{k}=0$$

  (d) The sequence $\left(a_{n}\right)_{n=0}^{\infty}$ is defined by $a_{0}=1$ and $a_{n+1}=2^{a_{n}}$ for $n \geqslant 0$. For any $k \in \mathbb{N}$, prove that

  $$\lim _{n \rightarrow \infty} \frac{2^{n^{k}}}{a_{n}}=0$$

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• Paper 1, Section I, $4 \mathrm{E}$

  #Analysis I

  2017

  Show that if the power series $\sum_{n=0}^{\infty} a_{n} z^{n}(z \in \mathbb{C})$ converges for some fixed $z=z_{0}$, then it converges absolutely for every $z$ satisfying $|z|<\left|z_{0}\right|$.

  Define the radius of convergence of a power series.

  Give an example of $v \in \mathbb{C}$ and an example of $w \in \mathbb{C}$ such that $|v|=|w|=1, \sum_{n=1}^{\infty} \frac{v^{n}}{n}$ converges and $\sum_{n=1}^{\infty} \frac{w^{n}}{n}$ diverges. [You may assume results about standard series without proof.] Use this to find the radius of convergence of the power series $\sum_{n=1}^{\infty} \frac{z^{n}}{n}$.

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• Paper 1, Section II, D

  #Analysis I

  2017

  (a) State the Intermediate Value Theorem.

  (b) Define what it means for a function $f: \mathbb{R} \rightarrow \mathbb{R}$ to be differentiable at a point $a \in \mathbb{R}$. If $f$ is differentiable everywhere on $\mathbb{R}$, must $f^{\prime}$ be continuous everywhere? Justify your answer.

  State the Mean Value Theorem.

  (c) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable everywhere. Let $a, b \in \mathbb{R}$ with $a<b$.

  If $f^{\prime}(a) \leqslant y \leqslant f^{\prime}(b)$, prove that there exists $c \in[a, b]$ such that $f^{\prime}(c)=y$. [Hint: consider the function $g$ defined by

  $$g(x)=\frac{f(x)-f(a)}{x-a}$$

  if $x \neq a$ and $\left.g(a)=f^{\prime}(a) .\right]$

  If additionally $f(a) \leqslant 0 \leqslant f(b)$, deduce that there exists $d \in[a, b]$ such that $f^{\prime}(d)+f(d)=y$.

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• Paper 1, Section II, D

  #Analysis I

  2017

  Let $a, b \in \mathbb{R}$ with $a<b$ and let $f:(a, b) \rightarrow \mathbb{R}$.

  (a) Define what it means for $f$ to be continuous at $y_{0} \in(a, b)$.

  $f$ is said to have a local minimum at $c \in(a, b)$ if there is some $\varepsilon>0$ such that $f(c) \leqslant f(x)$ whenever $x \in(a, b)$ and $|x-c|<\varepsilon$.

  If $f$ has a local minimum at $c \in(a, b)$ and $f$ is differentiable at $c$, show that $f^{\prime}(c)=0$.

  (b) $f$ is said to be convex if

  $$f(\lambda x+(1-\lambda) y) \leqslant \lambda f(x)+(1-\lambda) f(y)$$

  for every $x, y \in(a, b)$ and $\lambda \in[0,1]$. If $f$ is convex, $r \in \mathbb{R}$ and $\left[y_{0}-|r|, y_{0}+|r|\right] \subset(a, b)$, prove that

  $$(1+\lambda) f\left(y_{0}\right)-\lambda f\left(y_{0}-r\right) \leqslant f\left(y_{0}+\lambda r\right) \leqslant(1-\lambda) f\left(y_{0}\right)+\lambda f\left(y_{0}+r\right)$$

  for every $\lambda \in[0,1]$.

  Deduce that if $f$ is convex then $f$ is continuous.

  If $f$ is convex and has a local minimum at $c \in(a, b)$, prove that $f$ has a global minimum at $c$, i.e., that $f(x) \geqslant f(c)$ for every $x \in(a, b)$. [Hint: argue by contradiction.] Must $f$ be differentiable at $c$ ? Justify your answer.

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• Paper 1, Section II, E

  #Analysis I

  2017

  Let $f:[a, b] \rightarrow \mathbb{R}$ be a bounded function defined on the closed, bounded interval $[a, b]$ of $\mathbb{R}$. Suppose that for every $\varepsilon>0$ there is a dissection $\mathcal{D}$ of $[a, b]$ such that $S_{\mathcal{D}}(f)-s_{\mathcal{D}}(f)<\varepsilon$, where $s_{\mathcal{D}}(f)$ and $S_{\mathcal{D}}(f)$ denote the lower and upper Riemann sums of $f$ for the dissection $\mathcal{D}$. Deduce that $f$ is Riemann integrable. [You may assume without proof that $s_{\mathcal{D}}(f) \leqslant S_{\mathcal{D}^{\prime}}(f)$ for all dissections $\mathcal{D}$ and $\mathcal{D}^{\prime}$ of $\left.[a, b] .\right]$

  Prove that if $f:[a, b] \rightarrow \mathbb{R}$ is continuous, then $f$ is Riemann integrable.

  Let $g:(0,1] \rightarrow \mathbb{R}$ be a bounded continuous function. Show that for any $\lambda \in \mathbb{R}$, the function $f:[0,1] \rightarrow \mathbb{R}$ defined by

  $$f(x)= \begin{cases}g(x) & \text { if } 0<x \leqslant 1 \\ \lambda & \text { if } x=0\end{cases}$$

  is Riemann integrable.

  Let $f:[a, b] \rightarrow \mathbb{R}$ be a differentiable function with one-sided derivatives at the endpoints. Suppose that the derivative $f^{\prime}$ is (bounded and) Riemann integrable. Show that

  $$\int_{a}^{b} f^{\prime}(x) d x=f(b)-f(a)$$

  [You may use the Mean Value Theorem without proof.]

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• Paper 2, Section I, C

  #Differential Equations

  2017

  (a) The numbers $z_{1}, z_{2}, \ldots$ satisfy

  $$z_{n+1}=z_{n}+c_{n} \quad(n \geqslant 1),$$

  where $c_{1}, c_{2}, \ldots$ are given constants. Find $z_{n+1}$ in terms of $c_{1}, c_{2}, \ldots, c_{n}$ and $z_{1}$.

  (b) The numbers $x_{1}, x_{2}, \ldots$ satisfy

  $$x_{n+1}=a_{n} x_{n}+b_{n} \quad(n \geqslant 1),$$

  where $a_{1}, a_{2}, \ldots$ are given non-zero constants and $b_{1}, b_{2}, \ldots$ are given constants. Let $z_{1}=x_{1}$ and $z_{n+1}=x_{n+1} / U_{n}$, where $U_{n}=a_{1} a_{2} \cdots a_{n}$. Calculate $z_{n+1}-z_{n}$, and hence find $x_{n+1}$ in terms of $x_{1}, b_{1}, \ldots, b_{n}$ and $U_{1}, \ldots, U_{n}$.

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• Paper 2, Section I, $2 \mathrm{C}$

  #Differential Equations

  2017

  Consider the function

  $$f(x, y)=\frac{x}{y}+\frac{y}{x}-\frac{(x-y)^{2}}{a^{2}}$$

  defined for $x>0$ and $y>0$, where $a$ is a non-zero real constant. Show that $(\lambda, \lambda)$ is a stationary point of $f$ for each $\lambda>0$. Compute the Hessian and its eigenvalues at $(\lambda, \lambda)$.

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• Paper 2, Section II, C

  #Differential Equations

  2017

  The current $I(t)$ at time $t$ in an electrical circuit subject to an applied voltage $V(t)$ obeys the equation

  $$L \frac{d^{2} I}{d t^{2}}+R \frac{d I}{d t}+\frac{1}{C} I=\frac{d V}{d t}$$

  where $R, L$ and $C$ are the constant resistance, inductance and capacitance of the circuit with $R \geqslant 0, L>0$ and $C>0$.

  (a) In the case $R=0$ and $V(t)=0$, show that there exist time-periodic solutions of frequency $\omega_{0}$, which you should find.

  (b) In the case $V(t)=H(t)$, the Heaviside function, calculate, subject to the condition

  $$R^{2}>\frac{4 L}{C}$$

  the current for $t \geqslant 0$, assuming it is zero for $t<0$.

  (c) If $R>0$ and $V(t)=\sin \omega_{0} t$, where $\omega_{0}$ is as in part (a), show that there is a timeperiodic solution $I_{0}(t)$ of period $T=2 \pi / \omega_{0}$ and calculate its maximum value $I_{M}$.

  (i) Calculate the energy dissipated in each period, i.e., the quantity

  $$D=\int_{0}^{T} R I_{0}(t)^{2} d t$$

  Show that the quantity defined by

  $$Q=\frac{2 \pi}{D} \times \frac{L I_{M}^{2}}{2}$$

  satisfies $Q \omega_{0} R C=1$.

  (ii) Write down explicitly the general solution $I(t)$ for all $R>0$, and discuss the relevance of $I_{0}(t)$ to the large time behaviour of $I(t)$.

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• Paper 2, Section II, $7 \mathrm{C}$

  #Differential Equations

  2017

  Let $y_{1}$ and $y_{2}$ be two solutions of the differential equation

  $$y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0, \quad-\infty<x<\infty$$

  where $p$ and $q$ are given. Show, using the Wronskian, that

  • either there exist $\alpha$ and $\beta$, not both zero, such that $\alpha y_{1}(x)+\beta y_{2}(x)$ vanishes for all $x$,

  • or given $x_{0}, A$ and $B$, there exist $a$ and $b$ such that $y(x)=a y_{1}(x)+b y_{2}(x)$ satisfies the conditions $y\left(x_{0}\right)=A$ and $y^{\prime}\left(x_{0}\right)=B$.

  Find power series $y_{1}$ and $y_{2}$ such that an arbitrary solution of the equation

  $$y^{\prime \prime}(x)=x y(x)$$

  can be written as a linear combination of $y_{1}$ and $y_{2}$.

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• Paper 2, Section II, C

  #Differential Equations

  2017

  (a) Solve $\frac{d z}{d t}=z^{2}$ subject to $z(0)=z_{0}$. For which $z_{0}$ is the solution finite for all $t \in \mathbb{R}$ ?

  Let $a$ be a positive constant. By considering the lines $y=a\left(x-x_{0}\right)$ for constant $x_{0}$, or otherwise, show that any solution of the equation

  $$\frac{\partial f}{\partial x}+a \frac{\partial f}{\partial y}=0$$

  is of the form $f(x, y)=F(y-a x)$ for some function $F$.

  Solve the equation

  $$\frac{\partial f}{\partial x}+a \frac{\partial f}{\partial y}=f^{2}$$

  subject to $f(0, y)=g(y)$ for a given function $g$. For which $g$ is the solution bounded on $\mathbb{R}^{2}$ ?

  (b) By means of the change of variables $X=\alpha x+\beta y$ and $T=\gamma x+\delta y$ for appropriate real numbers $\alpha, \beta, \gamma, \delta$, show that the equation

  $$\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial x \partial y}=0$$

  can be transformed into the wave equation

  $$\frac{1}{c^{2}} \frac{\partial^{2} F}{\partial T^{2}}-\frac{\partial^{2} F}{\partial X^{2}}=0$$

  where $F$ is defined by $f(x, y)=F(\alpha x+\beta y, \gamma x+\delta y)$. Hence write down the general solution of $(*)$.

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• Paper 2, Section II, C

  #Differential Equations

  2017

  (a) Consider the system

  $$\begin{aligned} &\frac{d x}{d t}=x(1-x)-x y \\ &\frac{d y}{d t}=\frac{1}{8} y(4 x-1) \end{aligned}$$

  for $x(t) \geqslant 0, y(t) \geqslant 0$. Find the critical points, determine their type and explain, with the help of a diagram, the behaviour of solutions for large positive times $t$.

  (b) Consider the system

  $$\begin{aligned} &\frac{d x}{d t}=y+\left(1-x^{2}-y^{2}\right) x \\ &\frac{d y}{d t}=-x+\left(1-x^{2}-y^{2}\right) y \end{aligned}$$

  for $(x(t), y(t)) \in \mathbb{R}^{2}$. Rewrite the system in polar coordinates by setting $x(t)=$ $r(t) \cos \theta(t)$ and $y(t)=r(t) \sin \theta(t)$, and hence describe the behaviour of solutions for large positive and large negative times.

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• Paper 4, Section I, A

  #Dynamics and Relativity

  2017

  Consider a system of particles with masses $m_{i}$ and position vectors $\mathbf{x}_{i}$. Write down the definition of the position of the centre of mass $\mathbf{R}$ of the system. Let $T$ be the total kinetic energy of the system. Show that

  $$T=\frac{1}{2} M \dot{\mathbf{R}} \cdot \dot{\mathbf{R}}+\frac{1}{2} \sum_{i} m_{i} \dot{\mathbf{y}}_{i} \cdot \dot{\mathbf{y}}_{i}$$

  where $M$ is the total mass and $\mathbf{y}_{i}$ is the position vector of particle $i$ with respect to $\mathbf{R}$.

  The particles are connected to form a rigid body which rotates with angular speed $\omega$ about an axis $\mathbf{n}$ through $\mathbf{R}$, where $\mathbf{n} \cdot \mathbf{n}=1$. Show that

  $$T=\frac{1}{2} M \dot{\mathbf{R}} \cdot \dot{\mathbf{R}}+\frac{1}{2} I \omega^{2},$$

  where $I=\sum_{i} I_{i}$ and $I_{i}$ is the moment of inertia of particle $i$ about $\mathbf{n}$.

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• Paper 4, Section I, A

  #Dynamics and Relativity

  2017

  A tennis ball of mass $m$ is projected vertically upwards with initial speed $u_{0}$ and reaches its highest point at time $T$. In addition to uniform gravity, the ball experiences air resistance, which produces a frictional force of magnitude $\alpha v$, where $v$ is the ball's speed and $\alpha$ is a positive constant. Show by dimensional analysis that $T$ can be written in the form

  $$T=\frac{m}{\alpha} f(\lambda)$$

  for some function $f$ of a dimensionless quantity $\lambda$.

  Use the equation of motion of the ball to find $f(\lambda)$.

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• Paper 4, Section II, A

  #Dynamics and Relativity

  2017

  (a) A photon with energy $E_{1}$ in the laboratory frame collides with an electron of rest mass $m$ that is initially at rest in the laboratory frame. As a result of the collision the photon is deflected through an angle $\theta$ as measured in the laboratory frame and its energy changes to $E_{2}$.

  Derive an expression for $\frac{1}{E_{2}}-\frac{1}{E_{1}}$ in terms of $\theta, m$ and $c$.

  (b) A deuterium atom with rest mass $m_{1}$ and energy $E_{1}$ in the laboratory frame collides with another deuterium atom that is initially at rest in the laboratory frame. The result of this collision is a proton of rest mass $m_{2}$ and energy $E_{2}$, and a tritium atom of rest mass $m_{3}$. Show that, if the proton is emitted perpendicular to the incoming trajectory of the deuterium atom as measured in the laboratory frame, then

  $$m_{3}^{2}=m_{2}^{2}+2\left(m_{1}+\frac{E_{1}}{c^{2}}\right)\left(m_{1}-\frac{E_{2}}{c^{2}}\right)$$

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• Paper 4, Section II, A

  #Dynamics and Relativity

  2017

  A particle of unit mass moves under the influence of a central force. By considering the components of the acceleration in polar coordinates $(r, \theta)$ prove that the magnitude of the angular momentum is conserved. [You may use $\ddot{\mathbf{r}}=\left(\ddot{r}-r \dot{\theta}^{2}\right) \hat{\mathbf{r}}+(2 \dot{r} \dot{\theta}+r \ddot{\theta}) \hat{\boldsymbol{\theta}}$. ]

  Now suppose that the central force is derived from the potential $k / r$, where $k$ is a constant.

  (a) Show that the total energy of the particle can be written in the form

  $$E=\frac{1}{2} \dot{r}^{2}+V_{\mathrm{eff}}(r)$$

  Sketch $V_{\text {eff }}(r)$ in the cases $k>0$ and $k<0$.

  (b) The particle is projected from a very large distance from the origin with speed $v$ and impact parameter $b$. [The impact parameter is the distance of closest approach to the origin in absence of any force.]

  (i) In the case $k<0$, sketch the particle's trajectory and find the shortest distance $p$ between the particle and the origin, and the speed $u$ of the particle when $r=p$.

  (ii) In the case $k>0$, sketch the particle's trajectory and find the corresponding shortest distance $\widetilde{p}$ between the particle and the origin, and the speed $\widetilde{u}$ of the particle when $r=\widetilde{p}$.

  (iii) Find $p \tilde{p}$ and $u \tilde{u}$ in terms of $b$ and $v$. [In answering part (iii) you should assume that $|k|$ is the same in parts (i) and (ii).]

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• Paper 4, Section II, A

  #Dynamics and Relativity

  2017

  (a) Consider an inertial frame $S$, and a frame $S^{\prime}$ which rotates with constant angular velocity $\boldsymbol{\omega}$ relative to $S$. The two frames share a common origin. Identify each term in the equation

  $$\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S^{\prime}}=\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S}-2 \boldsymbol{\omega} \times\left(\frac{d \mathbf{r}}{d t}\right)_{S^{\prime}}-\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})$$

  (b) A small bead $P$ of unit mass can slide without friction on a circular hoop of radius $a$. The hoop is horizontal and rotating with constant angular speed $\omega$ about a fixed vertical axis through a point $O$ on its circumference.

  (i) Using Cartesian axes in the rotating frame $S^{\prime}$, with origin at $O$ and $x^{\prime}$-axis along the diameter of the hoop through $O$, write down the position vector of $P$ in terms of $a$ and the angle $\theta$ shown in the diagram $\left(-\frac{1}{2} \pi \leqslant \theta \leqslant \frac{1}{2} \pi\right)$.

  

  (ii) Working again in the rotating frame, find, in terms of $a, \theta, \dot{\theta}$ and $\omega$, an expression for the horizontal component of the force exerted by the hoop on the bead.

  (iii) For what value of $\theta$ is the bead in stable equilibrium? Find the frequency of small oscillations of the bead about that point.

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• Paper 4, Section II, A

  #Dynamics and Relativity

  2017

  (a) A rocket moves in a straight line with speed $v(t)$ and is subject to no external forces. The rocket is composed of a body of mass $M$ and fuel of mass $m(t)$, which is burnt at constant rate $\alpha$ and the exhaust is ejected with constant speed $u$ relative to the rocket. Show that

  $$(M+m) \frac{d v}{d t}-\alpha u=0 .$$

  Show that the speed of the rocket when all its fuel is burnt is

  $$v_{0}+u \log \left(1+\frac{m_{0}}{M}\right)$$

  where $v_{0}$ and $m_{0}$ are the speed of the rocket and the mass of the fuel at $t=0$.

  (b) A two-stage rocket moves in a straight line and is subject to no external forces. The rocket is initially at rest. The masses of the bodies of the two stages are $k M$ and $(1-k) M$, with $0 \leqslant k \leqslant 1$, and they initially carry masses $k m_{0}$ and $(1-k) m_{0}$ of fuel. Both stages burn fuel at a constant rate $\alpha$ when operating and the exhaust is ejected with constant speed $u$ relative to the rocket. The first stage operates first, until all its fuel is burnt. The body of the first stage is then detached with negligible force and the second stage ignites.

  Find the speed of the second stage when all its fuel is burnt. For $0 \leqslant k<1$ compare it with the speed of the rocket in part (a) in the case $v_{0}=0$. Comment on the case $k=1$.

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• Paper 3, Section I, E

  #Groups

  2017

  Let $w_{1}, w_{2}, w_{3}$ be distinct elements of $\mathbb{C} \cup\{\infty\}$. Write down the Möbius map $f$ that sends $w_{1}, w_{2}, w_{3}$ to $\infty, 0,1$, respectively. [Hint: You need to consider four cases.]

  Now let $w_{4}$ be another element of $\mathbb{C} \cup\{\infty\}$ distinct from $w_{1}, w_{2}, w_{3}$. Define the cross-ratio $\left[w_{1}, w_{2}, w_{3}, w_{4}\right]$ in terms of $f$.

  Prove that there is a circle or line through $w_{1}, w_{2}, w_{3}$ and $w_{4}$ if and only if the cross-ratio $\left[w_{1}, w_{2}, w_{3}, w_{4}\right]$ is real.

  [You may assume without proof that Möbius maps map circles and lines to circles and lines and also that there is a unique circle or line through any three distinct points of $\mathbb{C} \cup\{\infty\} .]$

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• Paper 3, Section I, E

  #Groups

  2017

  What does it mean to say that $H$ is a normal subgroup of the group $G$ ? For a normal subgroup $H$ of $G$ define the quotient group $G / H$. [You do not need to verify that $G / H$ is a group.]

  State the Isomorphism Theorem.

  Let

  $$G=\left\{\left(\begin{array}{ll} a & b \\ 0 & d \end{array}\right) \mid a, b, d \in \mathbb{R}, a d \neq 0\right\}$$

  be the group of $2 \times 2$ invertible upper-triangular real matrices. By considering a suitable homomorphism, show that the subset

  $$H=\left\{\left(\begin{array}{ll} 1 & b \\ 0 & 1 \end{array}\right) \mid b \in \mathbb{R}\right\}$$

  of $G$ is a normal subgroup of $G$ and identify the quotient $G / H$.

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• Paper 3, Section II, E

  #Groups

  2017

  Let $N$ be a normal subgroup of a finite group $G$ of prime index $p=|G: N|$.

  By considering a suitable homomorphism, show that if $H$ is a subgroup of $G$ that is not contained in $N$, then $H \cap N$ is a normal subgroup of $H$ of index $p$.

  Let $C$ be a conjugacy class of $G$ that is contained in $N$. Prove that $C$ is either a conjugacy class in $N$ or is the disjoint union of $p$ conjugacy classes in $N$.

  [You may use standard theorems without proof.]

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• Paper 3 , Section II, E

  #Groups

  2017

  State Lagrange's theorem. Show that the order of an element $x$ in a finite group $G$ is finite and divides the order of $G$.

  State Cauchy's theorem.

  List all groups of order 8 up to isomorphism. Carefully justify that the groups on your list are pairwise non-isomorphic and that any group of order 8 is isomorphic to one on your list. [You may use without proof the Direct Product Theorem and the description of standard groups in terms of generators satisfying certain relations.]

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• Paper 3, Section II, $7 \mathrm{E}$

  #Groups

  2017

  (a) Let $G$ be a finite group acting on a finite set $X$. State the Orbit-Stabiliser theorem. [Define the terms used.] Prove that

  $$\sum_{x \in X}|\operatorname{Stab}(x)|=n|G|$$

  where $n$ is the number of distinct orbits of $X$ under the action of $G$.

  Let $S=\{(g, x) \in G \times X: g \cdot x=x\}$, and for $g \in G$, let $\operatorname{Fix}(g)=\{x \in X: g \cdot x=x\}$.

  Show that

  $$|S|=\sum_{x \in X}|\operatorname{Stab}(x)|=\sum_{g \in G}|\operatorname{Fix}(g)|$$

  and deduce that

  $$n=\frac{1}{|G|} \sum_{g \in G}|\operatorname{Fix}(g)|$$

  (b) Let $H$ be the group of rotational symmetries of the cube. Show that $H$ has 24 elements. [If your proof involves calculating stabilisers, then you must carefully verify such calculations.]

  Using $(*)$, find the number of distinct ways of colouring the faces of the cube red, green and blue, where two colourings are distinct if one cannot be obtained from the other by a rotation of the cube. [A colouring need not use all three colours.]

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• Paper 3, Section II, E

  #Groups

  2017

  Prove that every element of the symmetric group $S_{n}$ is a product of transpositions. [You may assume without proof that every permutation is the product of disjoint cycles.]

  (a) Define the sign of a permutation in $S_{n}$, and prove that it is well defined. Define the alternating group $A_{n}$.

  (b) Show that $S_{n}$ is generated by the set $\left\{\left(\begin{array}{lll}1 & 2\end{array}\right),\left(\begin{array}{llll}1 & 2 & 3 & \ldots\end{array}\right)\right\}$.

  Given $1 \leqslant k<n$, prove that the set $\left\{\left(\begin{array}{ll}1 & \left.1+k),\left(\begin{array}{lll}1 & 2 & 3\end{array} \ldots n\right)\right\} \text { generates } S_{n} \text { if and only }\end{array}\right.\right.$ if $k$ and $n$ are coprime.

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• Paper 4, Section I, D

  #Numbers and Sets

  2017

  (a) Show that for all positive integers $z$ and $n$, either $z^{2 n} \equiv 0(\bmod 3)$ or $z^{2 n} \equiv 1(\bmod 3)$.

  (b) If the positive integers $x, y, z$ satisfy $x^{2}+y^{2}=z^{2}$, show that at least one of $x$ and $y$ must be divisible by 3 . Can both $x$ and $y$ be odd?

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• Paper 4, Section I, D

  #Numbers and Sets

  2017

  (a) Give the definitions of relation and equivalence relation on a set $S$.

  (b) Let $\Sigma$ be the set of ordered pairs $(A, f)$ where $A$ is a non-empty subset of $\mathbb{R}$ and $f: A \rightarrow \mathbb{R}$. Let $\mathcal{R}$ be the relation on $\Sigma$ defined by requiring $(A, f) \mathcal{R}(B, g)$ if the following two conditions hold:

  (i) $(A \backslash B) \cup(B \backslash A)$ is finite and

  (ii) there is a finite set $F \subset A \cap B$ such that $f(x)=g(x)$ for all $x \in A \cap B \backslash F$.

  Show that $\mathcal{R}$ is an equivalence relation on $\Sigma$.

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• Paper 4, Section II, D

  #Numbers and Sets

  2017

  (a) State and prove the Fermat-Euler Theorem. Deduce Fermat's Little Theorem. State Wilson's Theorem.

  (b) Let $p$ be an odd prime. Prove that $X^{2} \equiv-1(\bmod p)$ is solvable if and only if $p \equiv 1(\bmod 4)$.

  (c) Let $p$ be prime. If $h$ and $k$ are non-negative integers with $h+k=p-1$, prove that $h ! k !+(-1)^{h} \equiv 0(\bmod p) .$

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• Paper 4, Section II, D

  #Numbers and Sets

  2017

  (a) Define what it means for a set to be countable.

  (b) Let $A$ be an infinite subset of the set of natural numbers $\mathbb{N}=\{0,1,2, \ldots\}$. Prove that there is a bijection $f: \mathbb{N} \rightarrow A$.

  (c) Let $A_{n}$ be the set of natural numbers whose decimal representation ends with exactly $n-1$ zeros. For example, $71 \in A_{1}, 70 \in A_{2}$ and $15000 \in A_{4}$. By applying the result of part (b) with $A=A_{n}$, construct a bijection $g: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$. Deduce that the set of rationals is countable.

  (d) Let $A$ be an infinite set of positive real numbers. If every sequence $\left(a_{j}\right)_{j=1}^{\infty}$ of distinct elements with $a_{j} \in A$ for each $j$ has the property that

  $$\lim _{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^{N} a_{j}=0$$

  prove that $A$ is countable.

  [You may assume without proof that a countable union of countable sets is countable.]

  comment

• Paper 4, Section II, 7D

  #Numbers and Sets

  2017

  (a) For positive integers $n, m, k$ with $k \leqslant n$, show that

  $$\left(\begin{array}{l} n \\ k \end{array}\right)\left(\frac{k}{n}\right)^{m}=\left(\begin{array}{l} n-1 \\ k-1 \end{array}\right) \sum_{\ell=0}^{m-1} a_{n, m, \ell}\left(\frac{k-1}{n-1}\right)^{m-1-\ell}$$

  giving an explicit formula for $a_{n, m, \ell}$. [You may wish to consider the expansion of $\left.\left(\frac{k-1}{n-1}+\frac{1}{n-1}\right)^{m-1} .\right]$

  (b) For a function $f:[0,1] \rightarrow \mathbb{R}$ and each integer $n \geqslant 1$, the function $B_{n}(f):[0,1] \rightarrow \mathbb{R}$ is defined by

  $$B_{n}(f)(x)=\sum_{k=0}^{n} f\left(\frac{k}{n}\right)\left(\begin{array}{l} n \\ k \end{array}\right) x^{k}(1-x)^{n-k}$$

  For any integer $m \geqslant 0$ let $f_{m}(x)=x^{m}$. Show that $B_{n}\left(f_{0}\right)(x)=1$ and $B_{n}\left(f_{1}\right)(x)=x$ for all $n \geqslant 1$ and $x \in[0,1]$.

  Show that for each integer $m \geqslant 0$ and each $x \in[0,1]$,

  $$B_{n}\left(f_{m}\right)(x) \rightarrow f_{m}(x) \text { as } n \rightarrow \infty$$

  Deduce that for each integer $m \geqslant 0$,

  $$\lim _{n \rightarrow \infty} \frac{1}{4^{n}} \sum_{k=0}^{2 n}\left(\frac{k}{n}\right)^{m}\left(\begin{array}{c} 2 n \\ k \end{array}\right)=1$$

  comment

• Paper 4, Section II, D

  #Numbers and Sets

  2017

  Let $\left(a_{k}\right)_{k=1}^{\infty}$ be a sequence of real numbers.

  (a) Define what it means for $\left(a_{k}\right)_{k=1}^{\infty}$ to converge. Define what it means for the series $\sum_{k=1}^{\infty} a_{k}$ to converge.

  Show that if $\sum_{k=1}^{\infty} a_{k}$ converges, then $\left(a_{k}\right)_{k=1}^{\infty}$ converges to 0 .

  If $\left(a_{k}\right)_{k=1}^{\infty}$ converges to $a \in \mathbb{R}$, show that

  $$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} a_{k}=a$$

  (b) Suppose $a_{k}>0$ for every $k$. Let $u_{n}=\sum_{k=1}^{n}\left(a_{k}+\frac{1}{a_{k}}\right)$ and $v_{n}=\sum_{k=1}^{n}\left(a_{k}-\frac{1}{a_{k}}\right)$.

  Show that $\left(u_{n}\right)_{n=1}^{\infty}$ does not converge.

  Give an example of a sequence $\left(a_{k}\right)_{k=1}^{\infty}$ with $a_{k}>0$ and $a_{k} \neq 1$ for every $k$ such that $\left(v_{n}\right)_{n=1}^{\infty}$ converges.

  If $\left(v_{n}\right)_{n=1}^{\infty}$ converges, show that $\frac{u_{n}}{n} \rightarrow 2$.

  comment

• Paper 2, Section I, F

  #Probability

  2017

  Let $X$ be a non-negative integer-valued random variable such that $0<\mathbb{E}\left(X^{2}\right)<\infty$.

  Prove that

  $$\frac{\mathbb{E}(X)^{2}}{\mathbb{E}\left(X^{2}\right)} \leqslant \mathbb{P}(X>0) \leqslant \mathbb{E}(X)$$

  [You may use any standard inequality.]

  comment

• Paper 2, Section I, F

  #Probability

  2017

  Let $X$ and $Y$ be real-valued random variables with joint density function

  $$f(x, y)= \begin{cases}x e^{-x(y+1)} & \text { if } x \geqslant 0 \text { and } y \geqslant 0 \\ 0 & \text { otherwise. }\end{cases}$$

  (i) Find the conditional probability density function of $Y$ given $X$.

  (ii) Find the expectation of $Y$ given $X$.

  comment

• Paper 2, Section II, F

  #Probability

  2017

  For a positive integer $N, p \in[0,1]$, and $k \in\{0,1, \ldots, N\}$, let

  $$p_{k}(N, p)=\left(\begin{array}{c} N \\ k \end{array}\right) p^{k}(1-p)^{N-k}$$

  (a) For fixed $N$ and $p$, show that $p_{k}(N, p)$ is a probability mass function on $\{0,1, \ldots, N\}$ and that the corresponding probability distribution has mean $N p$ and variance $N p(1-p)$.

  (b) Let $\lambda>0$. Show that, for any $k \in\{0,1,2, \ldots\}$,

  $$\lim _{N \rightarrow \infty} p_{k}(N, \lambda / N)=\frac{e^{-\lambda} \lambda^{k}}{k !}$$

  Show that the right-hand side of $(*)$ is a probability mass function on $\{0,1,2, \ldots\}$.

  (c) Let $p \in(0,1)$ and let $a, b \in \mathbb{R}$ with $a<b$. For all $N$, find integers $k_{a}(N)$ and $k_{b}(N)$ such that

  $$\sum_{k=k_{a}(N)}^{k_{b}(N)} p_{k}(N, p) \rightarrow \frac{1}{\sqrt{2 \pi}} \int_{a}^{b} e^{-\frac{1}{2} x^{2}} d x \quad \text { as } N \rightarrow \infty$$

  [You may use the Central Limit Theorem.]

  comment

• Paper 2, Section II, 10F

  #Probability

  2017

  (a) For any random variable $X$ and $\lambda>0$ and $t>0$, show that

  $$\mathbb{P}(X>t) \leqslant \mathbb{E}\left(e^{\lambda X}\right) e^{-\lambda t}$$

  For a standard normal random variable $X$, compute $\mathbb{E}\left(e^{\lambda X}\right)$ and deduce that

  $$\mathbb{P}(X>t) \leqslant e^{-\frac{1}{2} t^{2}}$$

  (b) Let $\mu, \lambda>0, \mu \neq \lambda$. For independent random variables $X$ and $Y$ with distributions $\operatorname{Exp}(\lambda)$ and $\operatorname{Exp}(\mu)$, respectively, compute the probability density functions of $X+Y$ and $\min \{X, Y\}$.

  comment

• Paper 2, Section II, F

  #Probability

  2017

  Let $\beta>0$. The Curie-Weiss Model of ferromagnetism is the probability distribution defined as follows. For $n \in \mathbb{N}$, define random variables $S_{1}, \ldots, S_{n}$ with values in $\{\pm 1\}$ such that the probabilities are given by

  $$\mathbb{P}\left(S_{1}=s_{1}, \ldots, S_{n}=s_{n}\right)=\frac{1}{Z_{n, \beta}} \exp \left(\frac{\beta}{2 n} \sum_{i=1}^{n} \sum_{j=1}^{n} s_{i} s_{j}\right)$$

  where $Z_{n, \beta}$ is the normalisation constant

  $$Z_{n, \beta}=\sum_{s_{1} \in\{\pm 1\}} \cdots \sum_{s_{n} \in\{\pm 1\}} \exp \left(\frac{\beta}{2 n} \sum_{i=1}^{n} \sum_{j=1}^{n} s_{i} s_{j}\right)$$

  (a) Show that $\mathbb{E}\left(S_{i}\right)=0$ for any $i$.

  (b) Show that $\mathbb{P}\left(S_{2}=+1 \mid S_{1}=+1\right) \geqslant \mathbb{P}\left(S_{2}=+1\right)$. [You may use $\mathbb{E}\left(S_{i} S_{j}\right) \geqslant 0$ for all $i, j$ without proof. ]

  (c) Let $M=\frac{1}{n} \sum_{i=1}^{n} S_{i}$. Show that $M$ takes values in $E_{n}=\left\{-1+\frac{2 k}{n}: k=0, \ldots, n\right\}$, and that for each $m \in E_{n}$ the number of possible values of $\left(S_{1}, \ldots, S_{n}\right)$ such that $M=m$ is

  $$\frac{n !}{\left(\frac{1+m}{2} n\right) !\left(\frac{1-m}{2} n\right) !}$$

  Find $\mathbb{P}(M=m)$ for any $m \in E_{n}$.

  comment

• Paper 2, Section II, 12F

  #Probability

  2017

  (a) Let $k \in\{1,2, \ldots\}$. For $j \in\{0, \ldots, k+1\}$, let $D_{j}$ be the first time at which a simple symmetric random walk on $\mathbb{Z}$ with initial position $j$ at time 0 hits 0 or $k+1$. Show $\mathbb{E}\left(D_{j}\right)=j(k+1-j)$. [If you use a recursion relation, you do not need to prove that its solution is unique.]

  (b) Let $\left(S_{n}\right)$ be a simple symmetric random walk on $\mathbb{Z}$ starting at 0 at time $n=0$. For $k \in\{1,2, \ldots\}$, let $T_{k}$ be the first time at which $\left(S_{n}\right)$ has visited $k$ distinct vertices. In particular, $T_{1}=0$. Show $\mathbb{E}\left(T_{k+1}-T_{k}\right)=k$ for $k \geqslant 1$. [You may use without proof that, conditional on $S_{T_{k}}=i$, the random variables $\left(S_{T_{k}+n}\right)_{n \geqslant 0}$ have the distribution of a simple symmetric random walk starting at $i$.]

  (c) For $n \geqslant 3$, let $\mathbb{Z}_{n}$ be the circle graph consisting of vertices $0, \ldots, n-1$ and edges between $k$ and $k+1$ where $n$ is identified with 0 . Let $\left(Y_{i}\right)$ be a simple random walk on $\mathbb{Z}_{n}$ starting at time 0 from 0 . Thus $Y_{0}=0$ and conditional on $Y_{i}$ the random variable $Y_{i+1}$ is $Y_{i} \pm 1$ with equal probability (identifying $k+n$ with $k$ ).

  The cover time $T$ of the simple random walk on $\mathbb{Z}_{n}$ is the first time at which the random walk has visited all vertices. Show that $\mathbb{E}(T)=n(n-1) / 2$.

  comment

• Paper 3, Section I, B

  #Vector Calculus

  2017

  Use the change of variables $x=r \cosh \theta, y=r \sinh \theta$ to evaluate

  $$\int_{A} y d x d y$$

  where $A$ is the region of the $x y$-plane bounded by the two line segments:

  $$\begin{aligned} &y=0, \quad 0 \leqslant x \leqslant 1 \\ &5 y=3 x, \quad 0 \leqslant x \leqslant \frac{5}{4} \end{aligned}$$

  and the curve

  $$x^{2}-y^{2}=1, \quad x \geqslant 1$$

  comment

• Paper 3, Section $I$, B

  #Vector Calculus

  2017

  (a) The two sets of basis vectors $\mathbf{e}_{i}$ and $\mathbf{e}_{i}^{\prime}$ (where $i=1,2,3$ ) are related by

  $$\mathbf{e}_{i}^{\prime}=R_{i j} \mathbf{e}_{j}$$

  where $R_{i j}$ are the entries of a rotation matrix. The components of a vector $\mathbf{v}$ with respect to the two bases are given by

  $$\mathbf{v}=v_{i} \mathbf{e}_{i}=v_{i}^{\prime} \mathbf{e}_{i}^{\prime}$$

  Derive the relationship between $v_{i}$ and $v_{i}^{\prime}$.

  (b) Let $\mathbf{T}$ be a $3 \times 3$ array defined in each (right-handed orthonormal) basis. Using part (a), state and prove the quotient theorem as applied to $\mathbf{T}$.

  comment

• Paper 3, Section II, B

  #Vector Calculus

  2017

  (a) The time-dependent vector field $\mathbf{F}$ is related to the vector field $\mathbf{B}$ by

  $$\mathbf{F}(\mathbf{x}, t)=\mathbf{B}(\mathbf{z})$$

  where $\mathbf{z}=t \mathbf{x}$. Show that

  $$(\mathbf{x} \cdot \nabla) \mathbf{F}=t \frac{\partial \mathbf{F}}{\partial t} \text {. }$$

  (b) The vector fields $\mathbf{B}$ and $\mathbf{A}$ satisfy $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. Show that $\boldsymbol{\nabla} \cdot \mathbf{B}=0$.

  (c) The vector field $\mathbf{B}$ satisfies $\boldsymbol{\nabla} \cdot \mathbf{B}=0$. Show that

  $$\mathbf{B}(\mathbf{x})=\nabla \times(\mathbf{D}(\mathbf{x}) \times \mathbf{x})$$

  where

  $$\mathbf{D}(\mathbf{x})=\int_{0}^{1} t \mathbf{B}(t \mathbf{x}) d t$$

  comment

• Paper 3, Section II, B

  #Vector Calculus

  2017

  By a suitable choice of $\mathbf{u}$ in the divergence theorem

  $$\int_{V} \nabla \cdot \mathbf{u} d V=\int_{S} \mathbf{u} \cdot d \mathbf{S}$$

  show that

  $$\int_{V} \nabla \phi d V=\int_{S} \phi d \mathbf{S}$$

  for any continuously differentiable function $\phi$.

  For the curved surface of the cone

  $$\mathbf{x}=(r \cos \theta, r \sin \theta, \sqrt{3} r), \quad 0 \leqslant \sqrt{3} r \leqslant 1, \quad 0 \leqslant \theta \leqslant 2 \pi$$

  show that $d \mathbf{S}=(\sqrt{3} \cos \theta, \sqrt{3} \sin \theta,-1) r d r d \theta$.

  Verify that $(*)$ holds for this cone and $\phi(x, y, z)=z^{2}$.

  comment

• Paper 3, Section II, B

  #Vector Calculus

  2017

  (a) Let $\mathbf{x}=\mathbf{r}(s)$ be a smooth curve parametrised by arc length $s$. Explain the meaning of the terms in the equation

  $$\frac{d \mathbf{t}}{d s}=\kappa \mathbf{n},$$

  where $\kappa(s)$ is the curvature of the curve.

  Now let $\mathbf{b}=\mathbf{t} \times \mathbf{n}$. Show that there is a scalar $\tau(s)$ (the torsion) such that

  $$\frac{d \mathbf{b}}{d s}=-\tau \mathbf{n}$$

  and derive an expression involving $\kappa$ and $\tau$ for $\frac{d \mathbf{n}}{d s}$.

  (b) Given a (nowhere zero) vector field $\mathbf{F}$, the field lines, or integral curves, of $\mathbf{F}$ are the curves parallel to $\mathbf{F}(\mathbf{x})$ at each point $\mathbf{x}$. Show that the curvature $\kappa$ of the field lines of $\mathbf{F}$ satisfies

  $$\frac{\mathbf{F} \times(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{F}}{F^{3}}=\pm \kappa \mathbf{b}$$

  where $F=|\mathbf{F}|$.

  (c) Use $(*)$ to find an expression for the curvature at the point $(x, y, z)$ of the field lines of $\mathbf{F}(x, y, z)=(x, y,-z)$.

  comment

• Paper 3, Section II, B

  #Vector Calculus

  2017

  Let $S$ be a piecewise smooth closed surface in $\mathbb{R}^{3}$ which is the boundary of a volume $V$.

  (a) The smooth functions $\phi$ and $\phi_{1}$ defined on $\mathbb{R}^{3}$ satisfy

  $$\nabla^{2} \phi=\nabla^{2} \phi_{1}=0$$

  in $V$ and $\phi(\mathbf{x})=\phi_{1}(\mathbf{x})=f(\mathbf{x})$ on $S$. By considering an integral of $\boldsymbol{\nabla} \psi \cdot \boldsymbol{\nabla} \psi$, where $\psi=\phi-\phi_{1}$, show that $\phi_{1}=\phi$.

  (b) The smooth function $u$ defined on $\mathbb{R}^{3}$ satisfies $u(\mathbf{x})=f(\mathbf{x})+C$ on $S$, where $f$ is the function in part (a) and $C$ is constant. Show that

  $$\int_{V} \nabla u \cdot \nabla u d V \geqslant \int_{V} \nabla \phi \cdot \nabla \phi d V$$

  where $\phi$ is the function in part (a). When does equality hold?

  (c) The smooth function $w(\mathbf{x}, t)$ satisfies

  $$\nabla^{2} w=\frac{\partial w}{\partial t}$$

  in $V$ and $\frac{\partial w}{\partial t}=0$ on $S$ for all $t$. Show that

  $$\frac{d}{d t} \int_{V} \nabla w \cdot \nabla w d V \leqslant 0$$

  with equality only if $\nabla^{2} w=0$ in $V$.

  comment

• Paper 1, Section I, A

  #Vectors and Matrices

  2017

  Consider $z \in \mathbb{C}$ with $|z|=1$ and $\arg z=\theta$, where $\theta \in[0, \pi)$.

  (a) Prove algebraically that the modulus of $1+z$ is $2 \cos \frac{1}{2} \theta$ and that the argument is $\frac{1}{2} \theta$. Obtain these results geometrically using the Argand diagram.

  (b) Obtain corresponding results algebraically and geometrically for $1-z$.

  comment

• Paper 1, Section I, C

  #Vectors and Matrices

  2017

  Let $A$ and $B$ be real $n \times n$ matrices.

  Show that $(A B)^{T}=B^{T} A^{T}$.

  For any square matrix, the matrix exponential is defined by the series

  $$e^{A}=I+\sum_{k=1}^{\infty} \frac{A^{k}}{k !}$$

  Show that $\left(e^{A}\right)^{T}=e^{A^{T}}$. [You are not required to consider issues of convergence.]

  Calculate, in terms of $A$ and $A^{T}$, the matrices $Q_{0}, Q_{1}$ and $Q_{2}$ in the series for the matrix product

  $$e^{t A} e^{t A^{T}}=\sum_{k=0}^{\infty} Q_{k} t^{k}, \quad \text { where } t \in \mathbb{R}$$

  Hence obtain a relation between $A$ and $A^{T}$ which necessarily holds if $e^{t A}$ is an orthogonal matrix.

  comment

• Paper 1, Section II, A

  #Vectors and Matrices

  2017

  (a) Define the vector product $\mathbf{x} \times \mathbf{y}$ of the vectors $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^{3}$. Use suffix notation to prove that

  $$\mathbf{x} \times(\mathbf{x} \times \mathbf{y})=\mathbf{x}(\mathbf{x} \cdot \mathbf{y})-\mathbf{y}(\mathbf{x} \cdot \mathbf{x})$$

  (b) The vectors $\mathbf{x}_{n+1}(n=0,1,2, \ldots)$ are defined by $\mathbf{x}_{n+1}=\lambda \mathbf{a} \times \mathbf{x}_{n}$, where $\mathbf{a}$ and $\mathbf{x}_{0}$ are fixed vectors with $|\mathbf{a}|=1$ and $\mathbf{a} \times \mathbf{x}_{0} \neq \mathbf{0}$, and $\lambda$ is a positive constant.

  (i) Write $\mathbf{x}_{2}$ as a linear combination of $\mathbf{a}$ and $\mathbf{x}_{0}$. Further, for $n \geqslant 1$, express $\mathbf{x}_{n+2}$ in terms of $\lambda$ and $\mathbf{x}_{n}$. Show, for $n \geqslant 1$, that $\left|\mathbf{x}_{n}\right|=\lambda^{n}\left|\mathbf{a} \times \mathbf{x}_{0}\right|$.

  (ii) Let $X_{n}$ be the point with position vector $\mathbf{x}_{n}(n=0,1,2, \ldots)$. Show that $X_{1}, X_{2}, \ldots$ lie on a pair of straight lines.

  (iii) Show that the line segment $X_{n} X_{n+1}(n \geqslant 1)$ is perpendicular to $X_{n+1} X_{n+2}$. Deduce that $X_{n} X_{n+1}$ is parallel to $X_{n+2} X_{n+3}$.

  Show that $\mathbf{x}_{n} \rightarrow \mathbf{0}$ as $n \rightarrow \infty$ if $\lambda<1$, and give a sketch to illustrate the case $\lambda=1$.

  (iv) The straight line through the points $X_{n+1}$ and $X_{n+2}$ makes an angle $\theta$ with the straight line through the points $X_{n}$ and $X_{n+3}$. Find $\cos \theta$ in terms of $\lambda$.

  comment

• Paper 1, Section II, B

  #Vectors and Matrices

  2017

  (a) Show that the eigenvalues of any real $n \times n$ square matrix $A$ are the same as the eigenvalues of $A^{T}$.

  The eigenvalues of $A$ are $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$ and the eigenvalues of $A^{T} A$ are $\mu_{1}, \mu_{2}, \ldots$, $\mu_{n}$. Determine, by means of a proof or a counterexample, whether the following are necessary valid: (i) $\sum_{i=1}^{n} \mu_{i}=\sum_{i=1}^{n} \lambda_{i}^{2}$; (ii) $\prod_{i=1}^{n} \mu_{i}=\prod_{i=1}^{n} \lambda_{i}^{2}$.

  (b) The $3 \times 3$ matrix $B$ is given by

  $$B=I+\mathbf{m n}^{T}$$

  where $\mathbf{m}$ and $\mathbf{n}$ are orthogonal real unit vectors and $I$ is the $3 \times 3$ identity matrix.

  (i) Show that $\mathbf{m} \times \mathbf{n}$ is an eigenvector of $B$, and write down a linearly independent eigenvector. Find the eigenvalues of $B$ and determine whether $B$ is diagonalisable.

  (ii) Find the eigenvectors and eigenvalues of $B^{T} B$.

  comment

• Paper 1, Section II, B

  #Vectors and Matrices

  2017

  (a) Show that a square matrix $A$ is anti-symmetric if and only if $\mathbf{x}^{T} A \mathbf{x}=0$ for every vector $\mathbf{x}$.

  (b) Let $A$ be a real anti-symmetric $n \times n$ matrix. Show that the eigenvalues of $A$ are imaginary or zero, and that the eigenvectors corresponding to distinct eigenvalues are orthogonal (in the sense that $\mathbf{x}^{\dagger} \mathbf{y}=0$, where the dagger denotes the hermitian conjugate).

  (c) Let $A$ be a non-zero real $3 \times 3$ anti-symmetric matrix. Show that there is a real non-zero vector a such that $A \mathbf{a}=\mathbf{0}$.

  Now let $\mathbf{b}$ be a real vector orthogonal to $\mathbf{a}$. Show that $A^{2} \mathbf{b}=-\theta^{2} \mathbf{b}$ for some real number $\theta$.

  The matrix $e^{A}$ is defined by the exponential series $I+A+\frac{1}{2 !} A^{2}+\cdots$ Express $e^{A} \mathbf{a}$ and $e^{A} \mathbf{b}$ in terms of $\mathbf{a}, \mathbf{b}, A \mathbf{b}$ and $\theta$.

  [You are not required to consider issues of convergence.]

  comment

• Paper 1, Section II, $8 \mathrm{C}$

  #Vectors and Matrices

  2017

  (a) Given $\mathbf{y} \in \mathbb{R}^{3}$ consider the linear transformation $T$ which maps

  $$\mathbf{x} \mapsto T \mathbf{x}=\left(\mathbf{x} \cdot \mathbf{e}_{1}\right) \mathbf{e}_{1}+\mathbf{x} \times \mathbf{y}$$

  Express $T$ as a matrix with respect to the standard basis $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$, and determine the rank and the dimension of the kernel of $T$ for the cases (i) $\mathbf{y}=c_{1} \mathbf{e}_{1}$, where $c_{1}$ is a fixed number, and (ii) $\mathbf{y} \cdot \mathbf{e}_{1}=0$.

  (b) Given that the equation

  $$A B \mathbf{x}=\mathbf{d}$$

  where

  $$A=\left(\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 2 & 3 \\ 0 & 1 & 2 \end{array}\right), \quad B=\left(\begin{array}{ccc} 1 & 4 & 1 \\ -3 & -2 & 1 \\ 1 & -1 & -1 \end{array}\right) \quad \text { and } \quad \mathbf{d}=\left(\begin{array}{l} 1 \\ 1 \\ k \end{array}\right)$$

  has a solution, show that $4 k=1$.

  comment

• Paper 3, Section I, G

  #Analysis II

  2017

  What does it mean to say that a metric space is complete? Which of the following metric spaces are complete? Briefly justify your answers.

  (i) $[0,1]$ with the Euclidean metric.

  (ii) $\mathbb{Q}$ with the Euclidean metric.

  (iii) The subset

  $$\{(0,0)\} \cup\{(x, \sin (1 / x)) \mid x>0\} \subset \mathbb{R}^{2}$$

  with the metric induced from the Euclidean metric on $\mathbb{R}^{2}$.

  Write down a metric on $\mathbb{R}$ with respect to which $\mathbb{R}$ is not complete, justifying your answer.

  [You may assume throughout that $\mathbb{R}$ is complete with respect to the Euclidean metric.

  comment

• Paper 2, Section I, G

  #Analysis II

  2017

  Let $X \subset \mathbb{R}$. What does it mean to say that a sequence of real-valued functions on $X$ is uniformly convergent?

  Let $f, f_{n}(n \geqslant 1): \mathbb{R} \rightarrow \mathbb{R}$ be functions.

  (a) Show that if each $f_{n}$ is continuous, and $\left(f_{n}\right)$ converges uniformly on $\mathbb{R}$ to $f$, then $f$ is also continuous.

  (b) Suppose that, for every $M>0,\left(f_{n}\right)$ converges uniformly on $[-M, M]$. Need $\left(f_{n}\right)$ converge uniformly on $\mathbb{R}$ ? Justify your answer.

  comment

• Paper 4, Section I, G

  #Analysis II

  2017

  State the chain rule for the composition of two differentiable functions $f: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ and $g: \mathbb{R}^{n} \rightarrow \mathbb{R}^{p}$.

  Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be differentiable. For $c \in \mathbb{R}$, let $g(x)=f(x, c-x)$. Compute the derivative of $g$. Show that if $\partial f / \partial x=\partial f / \partial y$ throughout $\mathbb{R}^{2}$, then $f(x, y)=h(x+y)$ for some function $h: \mathbb{R} \rightarrow \mathbb{R}$.

  comment

• Paper 1, Section II, G

  #Analysis II

  2017

  What does it mean to say that a real-valued function on a metric space is uniformly continuous? Show that a continuous function on a closed interval in $\mathbb{R}$ is uniformly continuous.

  What does it mean to say that a real-valued function on a metric space is Lipschitz? Show that if a function is Lipschitz then it is uniformly continuous.

  Which of the following statements concerning continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ are true and which are false? Justify your answers.

  (i) If $f$ is bounded then $f$ is uniformly continuous.

  (ii) If $f$ is differentiable and $f^{\prime}$ is bounded, then $f$ is uniformly continuous.

  (iii) There exists a sequence of uniformly continuous functions converging pointwise to $f$.

  comment

• Paper 2, Section II, G

  #Analysis II

  2017

  Let $V$ be a real vector space. What is a norm on $V$ ? Show that if $\|-\|$ is a norm on $V$, then the maps $T_{v}: x \mapsto x+v\left(\right.$ for $v \in V$ ) and $m_{a}: x \mapsto a x$ (for $a \in \mathbb{R}$ ) are continuous with respect to the norm.

  Let $B \subset V$ be a subset containing 0 . Show that there exists at most one norm on $V$ for which $B$ is the open unit ball.

  Suppose that $B$ satisfies the following two properties:

  • if $v \in V$ is a nonzero vector, then the line $\mathbb{R} v \subset V$ meets $B$ in a set of the form $\{t v:-\lambda<t<\lambda\}$ for some $\lambda>0$;

  • if $x, y \in B$ and $s, t>0$ then $(s+t)^{-1}(s x+t y) \in B$.

  Show that there exists a norm $\|-\|_{B}$ for which $B$ is the open unit ball.

  Identify $\|-\|_{B}$ in the following two cases:

  (i) $V=\mathbb{R}^{n}, B=\left\{\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n}:-1<x_{i}<1\right.$ for all $\left.i\right\}$.

  (ii) $V=\mathbb{R}^{2}, B$ the interior of the square with vertices $(\pm 1,0),(0, \pm 1)$.

  Let $C \subset \mathbb{R}^{2}$ be the set

  $$C=\left\{\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2}:\left|x_{1}\right|<1,\left|x_{2}\right|<1, \text { and }\left(\left|x_{1}\right|-1\right)^{2}+\left(\left|x_{2}\right|-1\right)^{2}>1\right\}$$

  Is there a norm on $\mathbb{R}^{2}$ for which $C$ is the open unit ball? Justify your answer.

  comment

• Paper 4, Section II, G

  #Analysis II

  2017

  Let $U \subset \mathbb{R}^{m}$ be a nonempty open set. What does it mean to say that a function $f: U \rightarrow \mathbb{R}^{n}$ is differentiable?

  Let $f: U \rightarrow \mathbb{R}$ be a function, where $U \subset \mathbb{R}^{2}$ is open. Show that if the first partial derivatives of $f$ exist and are continuous on $U$, then $f$ is differentiable on $U$.

  Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be the function

  $$f(x, y)= \begin{cases}0 & (x, y)=(0,0) \\ \frac{x^{3}+2 y^{4}}{x^{2}+y^{2}} & (x, y) \neq(0,0)\end{cases}$$

  Determine, with proof, where $f$ is differentiable.

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• Paper 3, Section II, G

  #Analysis II

  2017

  What is a contraction map on a metric space $X$ ? State and prove the contraction mapping theorem.

  Let $(X, d)$ be a complete non-empty metric space. Show that if $f: X \rightarrow X$ is a map for which some iterate $f^{k}(k \geqslant 1)$ is a contraction map, then $f$ has a unique fixed point. Show that $f$ itself need not be a contraction map.

  Let $f:[0, \infty) \rightarrow[0, \infty)$ be the function

  $$f(x)=\frac{1}{3}\left(x+\sin x+\frac{1}{x+1}\right)$$

  Show that $f$ has a unique fixed point.

  comment

• Paper 4, Section I, 4F

  #Complex Analysis

  2017

  Let $D$ be a star-domain, and let $f$ be a continuous complex-valued function on $D$. Suppose that for every triangle $T$ contained in $D$ we have

  $$\int_{\partial T} f(z) d z=0$$

  Show that $f$ has an antiderivative on $D$.

  If we assume instead that $D$ is a domain (not necessarily a star-domain), does this conclusion still hold? Briefly justify your answer.

  comment

• Paper 2, Section II, 13A

  #Complex Analysis or Complex Methods

  2017

  State the residue theorem.

  By considering

  $$\oint_{C} \frac{z^{1 / 2} \log z}{1+z^{2}} d z$$

  with $C$ a suitably chosen contour in the upper half plane or otherwise, evaluate the real integrals

  $$\int_{0}^{\infty} \frac{x^{1 / 2} \log x}{1+x^{2}} d x$$

  and

  $$\int_{0}^{\infty} \frac{x^{1 / 2}}{1+x^{2}} d x$$

  where $x^{1 / 2}$ is taken to be the positive square root.

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• Paper 3, Section II, F

  #Complex Analysis

  2017

  Let $f$ be an entire function. Prove Taylor's theorem, that there exist complex numbers $c_{0}, c_{1}, \ldots$ such that $f(z)=\sum_{n=0}^{\infty} c_{n} z^{n}$ for all $z$. [You may assume Cauchy's Integral Formula.]

  For a positive real $r$, let $M_{r}=\sup \{|f(z)|:|z|=r\}$. Explain why we have

  $$\left|c_{n}\right| \leqslant \frac{M_{r}}{r^{n}}$$

  for all $n$.

  Now let $n$ and $r$ be fixed. For which entire functions $f$ do we have $\left|c_{n}\right|=\frac{M_{r}}{r^{n}} ?$

  comment

• Paper 1, Section I, A

  #Complex Analysis or Complex Methods

  2017

  Let $F(z)=u(x, y)+i v(x, y)$ where $z=x+i y$. Suppose $F(z)$ is an analytic function of $z$ in a domain $\mathcal{D}$ of the complex plane.

  Derive the Cauchy-Riemann equations satisfied by $u$ and $v$.

  For $u=\frac{x}{x^{2}+y^{2}}$ find a suitable function $v$ and domain $\mathcal{D}$ such that $F=u+i v$ is analytic in $\mathcal{D}$.

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• Paper 1, Section II, A

  #Complex Analysis or Complex Methods

  2017

  (a) Let $f(z)$ be defined on the complex plane such that $z f(z) \rightarrow 0$ as $|z| \rightarrow \infty$ and $f(z)$ is analytic on an open set containing $\operatorname{Im}(z) \geqslant-c$, where $c$ is a positive real constant.

  Let $C_{1}$ be the horizontal contour running from $-\infty-i c$ to $+\infty-i c$ and let

  $$F(\lambda)=\frac{1}{2 \pi i} \int_{C_{1}} \frac{f(z)}{z-\lambda} d z$$

  By evaluating the integral, show that $F(\lambda)$ is analytic for $\operatorname{Im}(\lambda)>-c$.

  (b) Let $g(z)$ be defined on the complex plane such that $z g(z) \rightarrow 0$ as $|z| \rightarrow \infty$ with $\operatorname{Im}(z) \geqslant-c$. Suppose $g(z)$ is analytic at all points except $z=\alpha_{+}$and $z=\alpha_{-}$which are simple poles with $\operatorname{Im}\left(\alpha_{+}\right)>c$ and $\operatorname{Im}\left(\alpha_{-}\right)<-c$.

  Let $C_{2}$ be the horizontal contour running from $-\infty+i c$ to $+\infty+i c$, and let

  $$\begin{gathered} H(\lambda)=\frac{1}{2 \pi i} \int_{C_{1}} \frac{g(z)}{z-\lambda} d z \\ J(\lambda)=-\frac{1}{2 \pi i} \int_{C_{2}} \frac{g(z)}{z-\lambda} d z . \end{gathered}$$

  (i) Show that $H(\lambda)$ is analytic for $\operatorname{Im}(\lambda)>-c$.

  (ii) Show that $J(\lambda)$ is analytic for $\operatorname{Im}(\lambda)<c$.

  (iii) Show that if $-c<\operatorname{Im}(\lambda)<c$ then $H(\lambda)+J(\lambda)=g(\lambda)$.

  [You should be careful to make sure you consider all points in the required regions.]

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• Paper 3, Section I, A

  #Complex Methods

  2017

  By using the Laplace transform, show that the solution to

  $$y^{\prime \prime}-4 y^{\prime}+3 y=t e^{-3 t},$$

  subject to the conditions $y(0)=0$ and $y^{\prime}(0)=1$, is given by

  $$y(t)=\frac{37}{72} e^{3 t}-\frac{17}{32} e^{t}+\left(\frac{5}{288}+\frac{1}{24} t\right) e^{-3 t}$$

  when $t \geqslant 0$.

  comment

• Paper 4, Section II, A

  #Complex Methods

  2017

  By using Fourier transforms and a conformal mapping

  $$w=\sin \left(\frac{\pi z}{a}\right)$$

  with $z=x+i y$ and $w=\xi+i \eta$, and a suitable real constant $a$, show that the solution to

  $$\begin{array}{rlrl} \nabla^{2} \phi & =0 & -2 \pi \leqslant x \leqslant 2 \pi, y \geqslant 0 \\ \phi(x, 0) & =f(x) & -2 \pi \leqslant x \leqslant 2 \pi \\ \phi(\pm 2 \pi, y) & =0 & y>0, \\ \phi(x, y) & \rightarrow 0 & y \rightarrow \infty,-2 \pi \leqslant x \leqslant 2 \pi \end{array}$$

  is given by

  $$\phi(\xi, \eta)=\frac{\eta}{\pi} \int_{-1}^{1} \frac{F\left(\xi^{\prime}\right)}{\eta^{2}+\left(\xi-\xi^{\prime}\right)^{2}} d \xi^{\prime}$$

  where $F\left(\xi^{\prime}\right)$ is to be determined.

  In the case of $f(x)=\sin \left(\frac{x}{4}\right)$, give $F\left(\xi^{\prime}\right)$ explicitly as a function of $\xi^{\prime}$. [You need not evaluate the integral.]

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• Paper 2, Section I, $\mathbf{6 C}$

  #Electromagnetism

  2017

  State Gauss's Law in the context of electrostatics.

  A spherically symmetric capacitor consists of two conductors in the form of concentric spherical shells of radii $a$ and $b$, with $b>a$. The inner sphere carries a charge $Q$ and the outer sphere carries a charge $-Q$. Determine the electric field $\mathbf{E}$ and the electrostatic potential $\phi$ in the regions $r<a, a<r<b$ and $r>b$. Show that the capacitance is

  $$C=\frac{4 \pi \epsilon_{0} a b}{b-a}$$

  and calculate the electrostatic energy of the system in terms of $Q$ and $C$.

  comment

• Paper 4 , Section I, $7 \mathrm{C}$

  #Electromagnetism

  2017

  A thin wire, in the form of a closed curve $C$, carries a constant current $I$. Using either the Biot-Savart law or the magnetic vector potential, show that the magnetic field far from the loop is of the approximate form

  $$\mathbf{B}(\mathbf{r}) \approx \frac{\mu_{0}}{4 \pi}\left[\frac{3(\mathbf{m} \cdot \mathbf{r}) \mathbf{r}-\mathbf{m}|\mathbf{r}|^{2}}{|\mathbf{r}|^{5}}\right]$$

  where $\mathbf{m}$ is the magnetic dipole moment of the loop. Derive an expression for $\mathbf{m}$ in terms of $I$ and the vector area spanned by the curve $C$.

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• Paper 1, Section II, C

  #Electromagnetism

  2017

  Write down Maxwell's equations for the electric field $\mathbf{E}(\mathbf{x}, t)$ and the magnetic field $\mathbf{B}(\mathbf{x}, t)$ in a vacuum. Deduce that both $\mathbf{E}$ and $\mathbf{B}$ satisfy a wave equation, and relate the wave speed $c$ to the physical constants $\epsilon_{0}$ and $\mu_{0}$.

  Verify that there exist plane-wave solutions of the form

  $$\begin{aligned} &\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{e} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right] \\ &\mathbf{B}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{b} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right] \end{aligned}$$

  where $\mathbf{e}$ and $\mathbf{b}$ are constant complex vectors, $\mathbf{k}$ is a constant real vector and $\omega$ is a real constant. Derive the dispersion relation that relates the angular frequency $\omega$ of the wave to the wavevector $\mathbf{k}$, and give the algebraic relations between the vectors $\mathbf{e}, \mathbf{b}$ and $\mathbf{k}$ implied by Maxwell's equations.

  Let $\mathbf{n}$ be a constant real unit vector. Suppose that a perfect conductor occupies the region $\mathbf{n} \cdot \mathbf{x}<0$ with a plane boundary $\mathbf{n} \cdot \mathbf{x}=0$. In the vacuum region $\mathbf{n} \cdot \mathbf{x}>0$, a plane electromagnetic wave of the above form, with $\mathbf{k} \cdot \mathbf{n}<0$, is incident on the plane boundary. Write down the boundary conditions on $\mathbf{E}$ and $\mathbf{B}$ at the surface of the conductor. Show that Maxwell's equations and the boundary conditions are satisfied if the solution in the vacuum region is the sum of the incident wave given above and a reflected wave of the form

  $$\begin{aligned} &\mathbf{E}^{\prime}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{e}^{\prime} e^{i\left(\mathbf{k}^{\prime} \cdot \mathbf{x}-\omega t\right)}\right] \\ &\mathbf{B}^{\prime}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{b}^{\prime} e^{i\left(\mathbf{k}^{\prime} \cdot \mathbf{x}-\omega t\right)}\right] \end{aligned}$$

  where

  $$\begin{aligned} &\mathbf{e}^{\prime}=-\mathbf{e}+2(\mathbf{n} \cdot \mathbf{e}) \mathbf{n} \\ &\mathbf{b}^{\prime}=\mathbf{b}-2(\mathbf{n} \cdot \mathbf{b}) \mathbf{n} \\ &\mathbf{k}^{\prime}=\mathbf{k}-2(\mathbf{n} \cdot \mathbf{k}) \mathbf{n} \end{aligned}$$

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• Paper 3, Section II, C

  #Electromagnetism

  2017

  (i) Two point charges, of opposite sign and unequal magnitude, are placed at two different locations. Show that the combined electrostatic potential vanishes on a sphere that encloses only the charge of smaller magnitude.

  (ii) A grounded, conducting sphere of radius $a$ is centred at the origin. A point charge $q$ is located outside the sphere at position vector $\mathbf{p}$. Formulate the differential equation and boundary conditions for the electrostatic potential outside the sphere. Using the result of part (i) or otherwise, show that the electric field outside the sphere is identical to that generated (in the absence of any conductors) by the point charge $q$ and an image charge $q^{\prime}$ located inside the sphere at position vector $\mathbf{p}^{\prime}$, provided that $\mathbf{p}^{\prime}$ and $q^{\prime}$ are chosen correctly.

  Calculate the magnitude and direction of the force experienced by the charge $q$.

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• Paper 2, Section II, C

  #Electromagnetism

  2017

  In special relativity, the electromagnetic fields can be derived from a 4-vector potential $A^{\mu}=(\phi / c, \mathbf{A})$. Using the Minkowski metric tensor $\eta_{\mu \nu}$ and its inverse $\eta^{\mu \nu}$, state how the electromagnetic tensor $F_{\mu \nu}$ is related to the 4-potential, and write out explicitly the components of both $F_{\mu \nu}$ and $F^{\mu \nu}$ in terms of those of $\mathbf{E}$ and $\mathbf{B}$.

  If $x^{\prime \mu}=\Lambda_{\nu}^{\mu} x^{\nu}$ is a Lorentz transformation of the spacetime coordinates from one inertial frame $\mathcal{S}$ to another inertial frame $\mathcal{S}^{\prime}$, state how $F^{\prime \mu \nu}$ is related to $F^{\mu \nu}$.

  Write down the Lorentz transformation matrix for a boost in standard configuration, such that frame $\mathcal{S}^{\prime}$ moves relative to frame $\mathcal{S}$ with speed $v$ in the $+x$ direction. Deduce the transformation laws

  $$\begin{aligned} E_{x}^{\prime} &=E_{x} \\ E_{y}^{\prime} &=\gamma\left(E_{y}-v B_{z}\right) \\ E_{z}^{\prime} &=\gamma\left(E_{z}+v B_{y}\right) \\ B_{x}^{\prime} &=B_{x} \\ B_{y}^{\prime} &=\gamma\left(B_{y}+\frac{v}{c^{2}} E_{z}\right) \\ B_{z}^{\prime} &=\gamma\left(B_{z}-\frac{v}{c^{2}} E_{y}\right) \end{aligned}$$

  where $\gamma=\left(1-\frac{v^{2}}{c^{2}}\right)^{-1 / 2}$

  In frame $\mathcal{S}$, an infinitely long wire of negligible thickness lies along the $x$ axis. The wire carries $n$ positive charges $+q$ per unit length, which travel at speed $u$ in the $+x$ direction, and $n$ negative charges $-q$ per unit length, which travel at speed $u$ in the $-x$ direction. There are no other sources of the electromagnetic field. Write down the electric and magnetic fields in $\mathcal{S}$ in terms of Cartesian coordinates. Calculate the electric field in frame $\mathcal{S}^{\prime}$, which is related to $\mathcal{S}$ by a boost by speed $v$ as described above. Give an explanation of the physical origin of your expression.

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• Paper 1, Section I, D

  #Fluid Dynamics

  2017

  For each of the flows

  (i) $\mathbf{u}=\left(2 x y, x^{2}+y^{2}\right)$

  (ii) $\mathbf{u}=(-2 y,-2 x)$

  determine whether or not the flow is incompressible and/or irrotational. Find the associated velocity potential and/or stream function when appropriate. For either one of the flows, sketch the streamlines of the flow, indicating the direction of the flow.

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• Paper 2, Section I, $7 \mathrm{D}$

  #Fluid Dynamics

  2017

  From Euler's equations describing steady inviscid fluid flow under the action of a conservative force, derive Bernoulli's equation for the pressure along a streamline of the flow, defining all variables that you introduce.

  Water fills an inverted, open, circular cone (radius increasing upwards) of half angle $\pi / 4$ to a height $h_{0}$ above its apex. At time $t=0$, the tip of the cone is removed to leave a small hole of radius $\epsilon \ll h_{0}$. Assuming that the flow is approximately steady while the depth of water $h(t)$ is much larger than $\epsilon$, show that the time taken for the water to drain is approximately

  $$\left(\frac{2}{25} \frac{h_{0}^{5}}{\epsilon^{4} g}\right)^{1 / 2}$$

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• Paper 1, Section II, D

  #Fluid Dynamics

  2017

  A layer of thickness $h$ of fluid of density $\rho$ and dynamic viscosity $\mu$ flows steadily down and parallel to a rigid plane inclined at angle $\alpha$ to the horizontal. Wind blows over the surface of the fluid and exerts a stress $S$ on the surface of the fluid in the upslope direction.

  (a) Draw a diagram of this situation, including indications of the applied stresses and body forces, a suitable coordinate system and a representation of the expected velocity profile.

  (b) Write down the equations and boundary conditions governing the flow, with a brief description of each, paying careful attention to signs. Solve these equations to determine the pressure and velocity fields.

  (c) Determine the volume flux and show that there is no net flux if

  $$S=\frac{2}{3} \rho g h \sin \alpha$$

  Draw a sketch of the corresponding velocity profile.

  (d) Determine the value of $S$ for which the shear stress on the rigid plane is zero and draw a sketch of the corresponding velocity profile.

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• Paper 4, Section II, D

  #Fluid Dynamics

  2017

  The linearised equations governing the horizontal components of flow $\mathbf{u}(x, y, t)$ in a rapidly rotating shallow layer of depth $h=h_{0}+\eta(x, y, t)$, where $\eta \ll h_{0}$, are

  $$\begin{gathered} \frac{\partial \mathbf{u}}{\partial t}+\mathbf{f} \times \mathbf{u}=-g \nabla \eta \\ \frac{\partial \eta}{\partial t}+h_{0} \nabla \cdot \mathbf{u}=0 \end{gathered}$$

  where $\mathbf{f}=f \mathbf{e}_{z}$ is the constant Coriolis parameter, and $\mathbf{e}_{z}$ is the unit vector in the vertical direction.

  Use these equations, either in vector form or using Cartesian components, to show that the potential vorticity

  $$\mathbf{Q}=\zeta-\frac{\eta}{h_{0}} \mathbf{f}$$

  is independent of time, where $\zeta=\nabla \times \mathbf{u}$ is the relative vorticity.

  Derive the equation

  $$\frac{\partial^{2} \eta}{\partial t^{2}}-g h_{0} \nabla^{2} \eta+f^{2} \eta=-h_{0} \mathbf{f} \cdot \mathbf{Q}$$

  In the case that $\mathbf{Q} \equiv 0$, determine and sketch the dispersion relation $\omega(k)$ for plane waves with $\eta=A e^{i(k x+\omega t)}$, where $A$ is constant. Discuss the nature of the waves qualitatively: do long waves propagate faster or slower than short waves; how does the phase speed depend on wavelength; does rotation have more effect on long waves or short waves; how dispersive are the waves?

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• Paper 3, Section II, D

  #Fluid Dynamics

  2017

  Use Euler's equations to derive the vorticity equation

  $$\frac{D \boldsymbol{\omega}}{D t}=\boldsymbol{\omega} \cdot \nabla \mathbf{u}$$

  where $\mathbf{u}$ is the fluid velocity and $\boldsymbol{\omega}$ is the vorticity.

  Consider axisymmetric, incompressible, inviscid flow between two rigid plates at $z=h(t)$ and $z=-h(t)$ in cylindrical polar coordinates $(r, \theta, z)$, where $t$ is time. Using mass conservation, or otherwise, find the complete flow field whose radial component is independent of $z$.

  Now suppose that the flow has angular velocity $\boldsymbol{\Omega}=\Omega(t) \mathbf{e}_{z}$ and that $\Omega=\Omega_{0}$ when $h=h_{0}$. Use the vorticity equation to determine the angular velocity for subsequent times as a function of $h$. What physical principle does your result illustrate?

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• Paper 1, Section I, G

  #Geometry

  2017

  Give the definition for the area of a hyperbolic triangle with interior angles $\alpha, \beta, \gamma$.

  Let $n \geqslant 3$. Show that the area of a convex hyperbolic $n$-gon with interior angles $\alpha_{1}, \ldots, \alpha_{n}$ is $(n-2) \pi-\sum \alpha_{i}$.

  Show that for every $n \geqslant 3$ and for every $A$ with $0<A<(n-2) \pi$ there exists a regular hyperbolic $n$-gon with area $A$.

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• Paper 3, Section I, G

  #Geometry

  2017

  Let

  $$\pi(x, y, z)=\frac{x+i y}{1-z}$$

  be stereographic projection from the unit sphere $S^{2}$ in $\mathbb{R}^{3}$ to the Riemann sphere $\mathbb{C}_{\infty}$. Show that if $r$ is a rotation of $S^{2}$, then $\pi r \pi^{-1}$ is a Möbius transformation of $\mathbb{C}_{\infty}$ which can be represented by an element of $S U(2)$. (You may assume without proof any result about generation of $S O(3)$ by a particular set of rotations, but should state it carefully.)

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• Paper 2, Section II, G

  #Geometry

  2017

  Let $H=\left\{\mathbf{x} \in \mathbb{R}^{n} \mid \mathbf{u} \cdot \mathbf{x}=c\right\}$ be a hyperplane in $\mathbb{R}^{n}$, where $\mathbf{u}$ is a unit vector and $c$ is a constant. Show that the reflection map

  $$\mathbf{x} \mapsto \mathbf{x}-2(\mathbf{u} \cdot \mathbf{x}-c) \mathbf{u}$$

  is an isometry of $\mathbb{R}^{n}$ which fixes $H$ pointwise.

  Let $\mathbf{p}, \mathbf{q}$ be distinct points in $\mathbb{R}^{n}$. Show that there is a unique reflection $R$ mapping $\mathbf{p}$ to $\mathbf{q}$, and that $R \in O(n)$ if and only if $\mathbf{p}$ and $\mathbf{q}$ are equidistant from the origin.

  Show that every isometry of $\mathbb{R}^{n}$ can be written as a product of at most $n+1$ reflections. Give an example of an isometry of $\mathbb{R}^{2}$ which cannot be written as a product of fewer than 3 reflections.

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• Paper 3, Section II, G

  #Geometry

  2017

  Let $\sigma: U \rightarrow \mathbb{R}^{3}$ be a parametrised surface, where $U \subset \mathbb{R}^{2}$ is an open set.

  (a) Explain what are the first and second fundamental forms of the surface, and what is its Gaussian curvature. Compute the Gaussian curvature of the hyperboloid $\sigma(x, y)=(x, y, x y)$.

  (b) Let $\mathbf{a}(x)$ and $\mathbf{b}(x)$ be parametrised curves in $\mathbb{R}^{3}$, and assume that

  $$\sigma(x, y)=\mathbf{a}(x)+y \mathbf{b}(x)$$

  Find a formula for the first fundamental form, and show that the Gaussian curvature vanishes if and only if

  $$\mathbf{a}^{\prime} \cdot\left(\mathbf{b} \times \mathbf{b}^{\prime}\right)=0$$

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• Paper 4, Section II, G

  #Geometry

  2017

  What is a hyperbolic line in (a) the disc model (b) the upper half-plane model of the hyperbolic plane? What is the hyperbolic distance $d(P, Q)$ between two points $P, Q$ in the hyperbolic plane? Show that if $\gamma$ is any continuously differentiable curve with endpoints $P$ and $Q$ then its length is at least $d(P, Q)$, with equality if and only if $\gamma$ is a monotonic reparametrisation of the hyperbolic line segment joining $P$ and $Q$.

  What does it mean to say that two hyperbolic lines $L, L^{\prime}$ are (a) parallel (b) ultraparallel? Show that $L$ and $L^{\prime}$ are ultraparallel if and only if they have a common perpendicular, and if so, then it is unique.

  A horocycle is a curve in the hyperbolic plane which in the disc model is a Euclidean circle with exactly one point on the boundary of the disc. Describe the horocycles in the upper half-plane model. Show that for any pair of horocycles there exists a hyperbolic line which meets both orthogonally. For which pairs of horocycles is this line unique?

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• Paper 3, Section I, E

  #Groups, Rings and Modules

  2017

  Let $R$ be a commutative ring and let $M$ be an $R$-module. Show that $M$ is a finitely generated $R$-module if and only if there exists a surjective $R$-module homomorphism $R^{n} \rightarrow M$ for some $n$.

  Find an example of a $\mathbb{Z}$-module $M$ such that there is no surjective $\mathbb{Z}$-module homomorphism $\mathbb{Z} \rightarrow M$ but there is a surjective $\mathbb{Z}$-module homomorphism $\mathbb{Z}^{2} \rightarrow M$ which is not an isomorphism. Justify your answer.

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• Paper 2, Section I, E

  #Groups, Rings and Modules

  2017

  (a) Define what is meant by a unique factorisation domain and by a principal ideal domain. State Gauss's lemma and Eisenstein's criterion, without proof.

  (b) Find an example, with justification, of a ring $R$ and a subring $S$ such that

  (i) $R$ is a principal ideal domain, and

  (ii) $S$ is a unique factorisation domain but not a principal ideal domain.

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• Paper 4, Section I, $2 E$

  #Groups, Rings and Modules

  2017

  Let $G$ be a non-trivial finite $p$-group and let $Z(G)$ be its centre. Show that $|Z(G)|>1$. Show that if $|G|=p^{3}$ and if $G$ is not abelian, then $|Z(G)|=p$.

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• Paper 1, Section II, 10E

  #Groups, Rings and Modules

  2017

  (a) State Sylow's theorem.

  (b) Let $G$ be a finite simple non-abelian group. Let $p$ be a prime number. Show that if $p$ divides $|G|$, then $|G|$ divides $n_{p} ! / 2$ where $n_{p}$ is the number of Sylow $p$-subgroups of $G$.

  (c) Let $G$ be a group of order 48 . Show that $G$ is not simple. Find an example of $G$ which has no normal Sylow 2-subgroup.

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• Paper 2, Section II, E

  #Groups, Rings and Modules

  2017

  Let $R$ be a commutative ring.

  (a) Let $N$ be the set of nilpotent elements of $R$, that is,

  $$N=\left\{r \in R \mid r^{n}=0 \text { for some } n \in \mathbb{N}\right\}$$

  Show that $N$ is an ideal of $R$.

  (b) Assume $R$ is Noetherian and assume $S \subset R$ is a non-empty subset such that if $s, t \in S$, then $s t \in S$. Let $I$ be an ideal of $R$ disjoint from $S$. Show that there is a prime ideal $P$ of $R$ containing $I$ and disjoint from $S$.

  (c) Again assume $R$ is Noetherian and let $N$ be as in part (a). Let $\mathcal{P}$ be the set of all prime ideals of $R$. Show that

  $$N=\bigcap_{P \in \mathcal{P}} P$$

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• Paper 4, Section II, E

  #Groups, Rings and Modules

  2017

  (a) State (without proof) the classification theorem for finitely generated modules over a Euclidean domain. Give the statement and the proof of the rational canonical form theorem.

  (b) Let $R$ be a principal ideal domain and let $M$ be an $R$-submodule of $R^{n}$. Show that $M$ is a free $R$-module.

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• Paper 3, Section II, E

  #Groups, Rings and Modules

  2017

  (a) Define what is meant by a Euclidean domain. Show that every Euclidean domain is a principal ideal domain.

  (b) Let $p \in \mathbb{Z}$ be a prime number and let $f \in \mathbb{Z}[x]$ be a monic polynomial of positive degree. Show that the quotient ring $\mathbb{Z}[x] /(p, f)$ is finite.

  (c) Let $\alpha \in \mathbb{Z}[\sqrt{-1}]$ and let $P$ be a non-zero prime ideal of $\mathbb{Z}[\alpha]$. Show that the quotient $\mathbb{Z}[\alpha] / P$ is a finite ring.

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• Paper 2, Section I, F

  #Linear Algebra

  2017

  State and prove the Rank-Nullity theorem.

  Let $\alpha$ be a linear map from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ of rank 2 . Give an example to show that $\mathbb{R}^{3}$ may be the direct sum of the kernel of $\alpha$ and the image of $\alpha$, and also an example where this is not the case.

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• Paper 1, Section I, F

  #Linear Algebra

  2017

  State and prove the Steinitz Exchange Lemma.

  Deduce that, for a subset $S$ of $\mathbb{R}^{n}$, any two of the following imply the third:

  (i) $S$ is linearly independent

  (ii) $S$ is spanning

  (iii) $S$ has exactly $n$ elements

  Let $e_{1}, e_{2}$ be a basis of $\mathbb{R}^{2}$. For which values of $\lambda$ do $\lambda e_{1}+e_{2}, e_{1}+\lambda e_{2}$ form a basis of $\mathbb{R}^{2} ?$

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• Paper 4, Section I, F

  #Linear Algebra

  2017

  Briefly explain the Gram-Schmidt orthogonalisation process in a real finite-dimensional inner product space $V$.

  For a subspace $U$ of $V$, define $U^{\perp}$, and show that $V=U \oplus U^{\perp}$.

  For which positive integers $n$ does

  $$(f, g)=f(1) g(1)+f(2) g(2)+f(3) g(3)$$

  define an inner product on the space of all real polynomials of degree at most $n$ ?

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• Paper 1, Section II, F

  #Linear Algebra

  2017

  Let $U$ and $V$ be finite-dimensional real vector spaces, and let $\alpha: U \rightarrow V$ be a surjective linear map. Which of the following are always true and which can be false? Give proofs or counterexamples as appropriate.

  (i) There is a linear map $\beta: V \rightarrow U$ such that $\beta \alpha$ is the identity map on $U$.

  (ii) There is a linear map $\beta: V \rightarrow U$ such that $\alpha \beta$ is the identity map on $V$.

  (iii) There is a subspace $W$ of $U$ such that the restriction of $\alpha$ to $W$ is an isomorphism from $W$ to $V$.

  (iv) If $X$ and $Y$ are subspaces of $U$ with $U=X \oplus Y$ then $V=\alpha(X) \oplus \alpha(Y)$.

  (v) If $X$ and $Y$ are subspaces of $U$ with $V=\alpha(X) \oplus \alpha(Y)$ then $U=X \oplus Y$.

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• Paper 2, Section II, F

  #Linear Algebra

  2017

  Let $\alpha: U \rightarrow V$ and $\beta: V \rightarrow W$ be linear maps between finite-dimensional real vector spaces.

  Show that the rank $r(\beta \alpha)$ satisfies $r(\beta \alpha) \leqslant \min (r(\beta), r(\alpha))$. Show also that $r(\beta \alpha) \geqslant r(\alpha)+r(\beta)-\operatorname{dim} V$. For each of these two inequalities, give examples to show that we may or may not have equality.

  Now let $V$ have dimension $2 n$ and let $\alpha: V \rightarrow V$ be a linear map of rank $2 n-2$ such that $\alpha^{n}=0$. Find the rank of $\alpha^{k}$ for each $1 \leqslant k \leqslant n-1$.

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• Paper 4, Section II, F

  #Linear Algebra

  2017

  What is the dual $X^{*}$ of a finite-dimensional real vector space $X$ ? If $X$ has a basis $e_{1}, \ldots, e_{n}$, define the dual basis, and prove that it is indeed a basis of $X^{*}$.

  [No results on the dimension of duals may be assumed without proof.]

  Write down (without making a choice of basis) an isomorphism from $X$ to $X^{* *}$. Prove that your map is indeed an isomorphism.

  Does every basis of $X^{*}$ arise as the dual basis of some basis of $X ?$ Justify your answer.

  A subspace $W$ of $X^{*}$ is called separating if for every non-zero $x \in X$ there is a $T \in W$ with $T(x) \neq 0$. Show that the only separating subspace of $X^{*}$ is $X^{*}$ itself.

  Now let $X$ be the (infinite-dimensional) space of all real polynomials. Explain briefly how we may identify $X^{*}$ with the space of all real sequences. Give an example of a proper subspace of $X^{*}$ that is separating.

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• Paper 3, Section II, F

  #Linear Algebra

  2017

  Let $f$ be a quadratic form on a finite-dimensional real vector space $V$. Prove that there exists a diagonal basis for $f$, meaning a basis with respect to which the matrix of $f$ is diagonal.

  Define the rank $r$ and signature $s$ of $f$ in terms of this matrix. Prove that $r$ and $s$ are independent of the choice of diagonal basis.

  In terms of $r, s$, and the dimension $n$ of $V$, what is the greatest dimension of a subspace on which $f$ is zero?

  Now let $f$ be the quadratic form on $\mathbb{R}^{3}$ given by $f(x, y, z)=x^{2}-y^{2}$. For which points $v$ in $\mathbb{R}^{3}$ is it the case that there is some diagonal basis for $f$ containing $v$ ?

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• Paper 3, Section I, H

  #Markov Chains

  2017

  (a) What does it mean to say that a Markov chain is reversible?

  (b) Let $G$ be a finite connected graph on $n$ vertices. What does it mean to say that $X$ is a simple random walk on $G$ ?

  Find the unique invariant distribution $\pi$ of $X$.

  Show that $X$ is reversible when $X_{0} \sim \pi$.

  [You may use, without proof, results about detailed balance equations, but you should state them clearly.]

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• Paper 4, Section I, $\mathbf{9 H}$

  #Markov Chains

  2017

  Prove that the simple symmetric random walk on $\mathbb{Z}^{3}$ is transient.

  [Any combinatorial inequality can be used without proof.]

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• Paper 1, Section II, H

  #Markov Chains

  2017

  A rich and generous man possesses $n$ pounds. Some poor cousins arrive at his mansion. Being generous he decides to give them money. On day 1 , he chooses uniformly at random an integer between $n-1$ and 1 inclusive and gives it to the first cousin. Then he is left with $x$ pounds. On day 2 , he chooses uniformly at random an integer between $x-1$ and 1 inclusive and gives it to the second cousin and so on. If $x=1$ then he does not give the next cousin any money. His choices of the uniform numbers are independent. Let $X_{i}$ be his fortune at the end of day $i$.

  Show that $X$ is a Markov chain and find its transition probabilities.

  Let $\tau$ be the first time he has 1 pound left, i.e. $\tau=\min \left\{i \geqslant 1: X_{i}=1\right\}$. Show that

  $$\mathbb{E}[\tau]=\sum_{i=1}^{n-1} \frac{1}{i}$$

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• Paper 2, Section II, H

  #Markov Chains

  2017

  Let $Y_{1}, Y_{2}, \ldots$ be i.i.d. random variables with values in $\{1,2, \ldots\}$ and $\mathbb{E}\left[Y_{1}\right]=\mu<\infty$. Moreover, suppose that the greatest common divisor of $\left\{n: \mathbb{P}\left(Y_{1}=n\right)>0\right\}$ is 1 . Consider the following process

  $$X_{n}=\inf \left\{m \geqslant n: Y_{1}+\ldots+Y_{k}=m, \text { for some } k \geqslant 0\right\}-n .$$

  (a) Show that $X$ is a Markov chain and find its transition probabilities.

  (b) Let $T_{0}=\inf \left\{n \geqslant 1: X_{n}=0\right\}$. Find $\mathbb{E}_{0}\left[T_{0}\right]$.

  (c) Find the limit as $n \rightarrow \infty$ of $\mathbb{P}\left(X_{n}=0\right)$. State carefully any theorems from the course that you are using.

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• Paper 2, Section I, B

  #Methods

  2017

  Expand $f(x)=x$ as a Fourier series on $-\pi<x<\pi$.

  By integrating the series show that $x^{2}$ on $-\pi<x<\pi$ can be written as

  $$x^{2}=\frac{a_{0}}{2}+\sum_{n=1}^{\infty} a_{n} \cos n x$$

  where $a_{n}, n=1,2, \ldots$, should be determined and

  $$a_{0}=8 \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}} .$$

  By evaluating $a_{0}$ another way show that

  $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}}=\frac{\pi^{2}}{12}$$

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• Paper 4, Section I, A

  #Methods

  2017

  The Legendre polynomials, $P_{n}(x)$ for integers $n \geqslant 0$, satisfy the Sturm-Liouville equation

  $$\frac{d}{d x}\left[\left(1-x^{2}\right) \frac{d}{d x} P_{n}(x)\right]+n(n+1) P_{n}(x)=0$$

  and the recursion formula

  $$(n+1) P_{n+1}(x)=(2 n+1) x P_{n}(x)-n P_{n-1}(x), \quad P_{0}(x)=1, \quad P_{1}(x)=x$$

  (i) For all $n \geqslant 0$, show that $P_{n}(x)$ is a polynomial of degree $n$ with $P_{n}(1)=1$.

  (ii) For all $m, n \geqslant 0$, show that $P_{n}(x)$ and $P_{m}(x)$ are orthogonal over the range $x \in[-1,1]$ when $m \neq n$.

  (iii) For each $n \geqslant 0$ let

  $$R_{n}(x)=\frac{d^{n}}{d x^{n}}\left[\left(x^{2}-1\right)^{n}\right]$$

  Assume that for each $n$ there is a constant $\alpha_{n}$ such that $P_{n}(x)=\alpha_{n} R_{n}(x)$ for all $x$. Determine $\alpha_{n}$ for each $n$.

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• Paper 3, Section I, A

  #Methods

  2017

  Using the substitution $u(x, y)=v(x, y) e^{-x^{2}}$, find $u(x, y)$ that satisfies

  $$u_{x}+x u_{y}+2 x u=e^{-x^{2}}$$

  with boundary data $u(0, y)=y e^{-y^{2}}$.

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• Paper 1, Section II, 14B

  #Methods

  2017

  (a)

  (i) Compute the Fourier transform $\tilde{h}(k)$ of $h(x)=e^{-a|x|}$, where $a$ is a real positive constant.

  (ii) Consider the boundary value problem

  $$-\frac{d^{2} u}{d x^{2}}+\omega^{2} u=e^{-|x|} \quad \text { on }-\infty<x<\infty$$

  with real constant $\omega \neq \pm 1$ and boundary condition $u(x) \rightarrow 0$ as $|x| \rightarrow \infty$.

  Find the Fourier transform $\tilde{u}(k)$ of $u(x)$ and hence solve the boundary value problem. You should clearly state any properties of the Fourier transform that you use.

  (b) Consider the wave equation

  $$v_{t t}=v_{x x} \quad \text { on } \quad-\infty<x<\infty \text { and } t>0$$

  with initial conditions

  $$v(x, 0)=f(x) \quad v_{t}(x, 0)=g(x) .$$

  Show that the Fourier transform $\tilde{v}(k, t)$ of the solution $v(x, t)$ with respect to the variable $x$ is given by

  $$\tilde{v}(k, t)=\tilde{f}(k) \cos k t+\frac{\tilde{g}(k)}{k} \sin k t$$

  where $\tilde{f}(k)$ and $\tilde{g}(k)$ are the Fourier transforms of the initial conditions. Starting from $\tilde{v}(k, t)$ derive d'Alembert's solution for the wave equation:

  $$v(x, t)=\frac{1}{2}(f(x-t)+f(x+t))+\frac{1}{2} \int_{x-t}^{x+t} g(\xi) d \xi$$

  You should state clearly any properties of the Fourier transform that you use.

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• Paper 3, Section II, A

  #Methods

  2017

  Let $\mathcal{L}$ be the linear differential operator

  $$\mathcal{L} y=y^{\prime \prime \prime}-y^{\prime \prime}-2 y^{\prime}$$

  where $^{\prime}$ denotes differentiation with respect to $x$.

  Find the Green's function, $G(x ; \xi)$, for $\mathcal{L}$ satisfying the homogeneous boundary conditions $G(0 ; \xi)=0, G^{\prime}(0 ; \xi)=0, G^{\prime \prime}(0 ; \xi)=0$.

  Using the Green's function, solve

  $$\mathcal{L} y=e^{x} \Theta(x-1)$$

  with boundary conditions $y(0)=1, y^{\prime}(0)=-1, y^{\prime \prime}(0)=0$. Here $\Theta(x)$ is the Heaviside step function having value 0 for $x<0$ and 1 for $x>0$.

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• Paper 2, Section II, A

  #Methods

  2017

  Laplace's equation for $\phi$ in cylindrical coordinates $(r, \theta, z)$, is

  $$\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}+\frac{\partial^{2} \phi}{\partial z^{2}}=0$$

  Use separation of variables to find an expression for the general solution to Laplace's equation in cylindrical coordinates that is $2 \pi$-periodic in $\theta$.

  Find the bounded solution $\phi(r, \theta, z)$ that satisfies

  $$\begin{aligned} \nabla^{2} \phi &=0 \quad z \geqslant 0, \quad 0 \leqslant r \leqslant 1 \\ \phi(1, \theta, z) &=e^{-4 z}(\cos \theta+\sin 2 \theta)+2 e^{-z} \sin 2 \theta \end{aligned}$$

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• Paper 4, Section II, B

  #Methods

  2017

  (a)

  (i) For the diffusion equation

  $$\frac{\partial \phi}{\partial t}-K \frac{\partial^{2} \phi}{\partial x^{2}}=0 \quad \text { on }-\infty<x<\infty \text { and } t \geqslant 0$$

  with diffusion constant $K$, state the properties (in terms of the Dirac delta function) that define the fundamental solution $F(x, t)$ and the Green's function $G(x, t ; y, \tau)$.

  You are not required to give expressions for these functions.

  (ii) Consider the initial value problem for the homogeneous equation:

  $$\frac{\partial \phi}{\partial t}-K \frac{\partial^{2} \phi}{\partial x^{2}}=0, \quad \phi\left(x, t_{0}\right)=\alpha(x) \quad \text { on }-\infty<x<\infty \text { and } t \geqslant t_{0}$$

  and the forced equation with homogeneous initial condition (and given forcing term $h(x, t))$ :

  $$\frac{\partial \phi}{\partial t}-K \frac{\partial^{2} \phi}{\partial x^{2}}=h(x, t), \quad \phi(x, 0)=0 \quad \text { on }-\infty<x<\infty \text { and } t \geqslant 0$$

  Given that $F$ and $G$ in part (i) are related by

  $$G(x, t ; y, \tau)=\Theta(t-\tau) F(x-y, t-\tau)$$

  (where $\Theta(t)$ is the Heaviside step function having value 0 for $t<0$ and 1 for $t>0$, show how the solution of (B) can be expressed in terms of solutions of (A) with suitable initial conditions. Briefly interpret your expression.

  (b) A semi-infinite conducting plate lies in the $\left(x_{1}, x_{2}\right)$ plane in the region $x_{1} \geqslant 0$. The boundary along the $x_{2}$ axis is perfectly insulated. Let $(r, \theta)$ denote standard polar coordinates on the plane. At time $t=0$ the entire plate is at temperature zero except for the region defined by $-\pi / 4<\theta<\pi / 4$ and $1<r<2$ which has constant initial temperature $T_{0}>0$. Subsequently the temperature of the plate obeys the two-dimensional heat equation with diffusion constant $K$. Given that the fundamental solution of the twodimensional heat equation on $\mathbb{R}^{2}$ is

  $$F\left(x_{1}, x_{2}, t\right)=\frac{1}{4 \pi K t} e^{-\left(x_{1}^{2}+x_{2}^{2}\right) /(4 K t)}$$

  show that the origin $(0,0)$ on the plate reaches its maximum temperature at time $t=3 /(8 K \log 2)$.

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• Paper 3, Section I, $3 E$

  #Metric and Topological Spaces

  2017

  Let $X$ and $Y$ be topological spaces.

  (a) Define what is meant by the product topology on $X \times Y$. Define the projection maps $p: X \times Y \rightarrow X$ and $q: X \times Y \rightarrow Y$ and show they are continuous.

  (b) Consider $\Delta=\{(x, x) \mid x \in X\}$ in $X \times X$. Show that $X$ is Hausdorff if and only if $\Delta$ is a closed subset of $X \times X$ in the product topology.

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• Paper 2, Section I, E

  #Metric and Topological Spaces

  2017

  Let $f:(X, d) \rightarrow(Y, e)$ be a function between metric spaces.

  (a) Give the $\epsilon-\delta$ definition for $f$ to be continuous. Show that $f$ is continuous if and only if $f^{-1}(U)$ is an open subset of $X$ for each open subset $U$ of $Y$.

  (b) Give an example of $f$ such that $f$ is not continuous but $f(V)$ is an open subset of $Y$ for every open subset $V$ of $X$.

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• Paper 1, Section II, E

  #Metric and Topological Spaces

  2017

  Consider $\mathbb{R}$ and $\mathbb{R}^{2}$ with their usual Euclidean topologies.

  (a) Show that a non-empty subset of $\mathbb{R}$ is connected if and only if it is an interval. Find a compact subset $K \subset \mathbb{R}$ for which $\mathbb{R} \backslash K$ has infinitely many connected components.

  (b) Let $T$ be a countable subset of $\mathbb{R}^{2}$. Show that $\mathbb{R}^{2} \backslash T$ is path-connected. Deduce that $\mathbb{R}^{2}$ is not homeomorphic to $\mathbb{R}$.

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• Paper 4, Section II, E

  #Metric and Topological Spaces

  2017

  Let $f: X \rightarrow Y$ be a continuous map between topological spaces.

  (a) Assume $X$ is compact and that $Z \subseteq X$ is a closed subset. Prove that $Z$ and $f(Z)$ are both compact.

  (b) Suppose that

  (i) $f^{-1}(\{y\})$ is compact for each $y \in Y$, and

  (ii) if $A$ is any closed subset of $X$, then $f(A)$ is a closed subset of $Y$.

  Show that if $K \subseteq Y$ is compact, then $f^{-1}(K)$ is compact.

  $\left[\right.$ Hint: Given an open cover $f^{-1}(K) \subseteq \bigcup_{i \in I} U_{i}$, find a finite subcover, say $f^{-1}(\{y\}) \subseteq$ $\bigcup_{i \in I_{y}} U_{i}$, for each $y \in K$; use closedness of $X \backslash \bigcup_{i \in I_{y}} U_{i}$ and property (ii) to produce an open cover of $K$.]

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• Paper 1, Section I, C

  #Numerical Analysis

  2017

  Given $n+1$ real points $x_{0}<x_{1}<\cdots<x_{n}$, define the Lagrange cardinal polynomials $\ell_{i}(x), i=0,1, \ldots, n$. Let $p(x)$ be the polynomial of degree $n$ that interpolates the function $f \in C^{n}\left[x_{0}, x_{n}\right]$ at these points. Express $p(x)$ in terms of the values $f_{i}=f\left(x_{i}\right)$, $i=0,1, \ldots, n$ and the Lagrange cardinal polynomials.

  Define the divided difference $f\left[x_{0}, x_{1}, \ldots, x_{n}\right]$ and give an expression for it in terms of $f_{0}, f_{1}, \ldots, f_{n}$ and $x_{0}, x_{1}, \ldots, x_{n}$. Prove that

  $$f\left[x_{0}, x_{1}, \ldots, x_{n}\right]=\frac{1}{n !} f^{(n)}(\xi)$$

  for some number $\xi \in\left[x_{0}, x_{n}\right]$.

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• Paper 4, Section I, C

  #Numerical Analysis

  2017

  For the matrix

  $$A=\left[\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & 5 & 5 & 5 \\ 1 & 5 & 14 & 14 \\ 1 & 5 & 14 & \lambda \end{array}\right]$$

  find a factorization of the form

  $$A=L D L^{\top} \text {, }$$

  where $D$ is diagonal and $L$ is lower triangular with ones on its diagonal.

  For what values of $\lambda$ is $A$ positive definite?

  In the case $\lambda=30$ find the Cholesky factorization of $A$.

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• Paper 1, Section II, C

  #Numerical Analysis

  2017

  A three-stage explicit Runge-Kutta method for solving the autonomous ordinary differential equation

  $$\frac{d y}{d t}=f(y)$$

  is given by

  $$y_{n+1}=y_{n}+h\left(b_{1} k_{1}+b_{2} k_{2}+b_{3} k_{3}\right),$$

  where

  $$\begin{aligned} &k_{1}=f\left(y_{n}\right) \\ &k_{2}=f\left(y_{n}+h a_{1} k_{1}\right) \\ &k_{3}=f\left(y_{n}+h\left(a_{2} k_{1}+a_{3} k_{2}\right)\right) \end{aligned}$$

  and $h>0$ is the time-step. Derive sufficient conditions on the coefficients $b_{1}, b_{2}, b_{3}, a_{1}$, $a_{2}$ and $a_{3}$ for the method to be of third order.

  Assuming that these conditions hold, verify that $-\frac{5}{2}$ belongs to the linear stability domain of the method.

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• Paper 2, Section II, C

  #Numerical Analysis

  2017

  Define the linear least-squares problem for the equation $A \mathbf{x}=\mathbf{b}$, where $A$ is an $m \times n$ matrix with $m>n, \mathbf{b} \in \mathbb{R}^{m}$ is a given vector and $\mathbf{x} \in \mathbb{R}^{n}$ is an unknown vector.

  If $A=Q R$, where $Q$ is an orthogonal matrix and $R$ is an upper triangular matrix in standard form, explain why the least-squares problem is solved by minimizing the Euclidean norm $\left\|R \mathbf{x}-Q^{\top} \mathbf{b}\right\|$.

  Using the method of Householder reflections, find a QR factorization of the matrix

  $$A=\left[\begin{array}{rrr} 1 & 3 & 3 \\ 1 & 3 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & -1 \end{array}\right]$$

  Hence find the solution of the least-squares problem in the case

  $$\mathbf{b}=\left[\begin{array}{r} 1 \\ 1 \\ 3 \\ -1 \end{array}\right]$$

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• Paper 3, Section II, C

  #Numerical Analysis

  2017

  Let $p_{n} \in \mathbb{P}_{n}$ be the $n$th monic orthogonal polynomial with respect to the inner product

  $$\langle f, g\rangle=\int_{a}^{b} w(x) f(x) g(x) d x$$

  on $C[a, b]$, where $w$ is a positive weight function.

  Prove that, for $n \geqslant 1, p_{n}$ has $n$ distinct zeros in the interval $(a, b)$.

  Let $c_{1}, c_{2}, \ldots, c_{n} \in[a, b]$ be $n$ distinct points. Show that the quadrature formula

  $$\int_{a}^{b} w(x) f(x) d x \approx \sum_{i=1}^{n} b_{i} f\left(c_{i}\right)$$

  is exact for all $f \in \mathbb{P}_{n-1}$ if the weights $b_{i}$ are chosen to be

  $$b_{i}=\int_{a}^{b} w(x) \prod_{\substack{j=1 \\ j \neq i}}^{n} \frac{x-c_{j}}{c_{i}-c_{j}} d x$$

  Show further that the quadrature formula is exact for all $f \in \mathbb{P}_{2 n-1}$ if the nodes $c_{i}$ are chosen to be the zeros of $p_{n}$ (Gaussian quadrature). [Hint: Write $f$ as $q p_{n}+r$, where $q, r \in \mathbb{P}_{n-1}$.]

  Use the Peano kernel theorem to write an integral expression for the approximation error of Gaussian quadrature for sufficiently differentiable functions. (You should give a formal expression for the Peano kernel but are not required to evaluate it.)

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• Paper 1, Section I, H

  #Optimization

  2017

  Solve the following linear programming problem using the simplex method:

  $$\begin{array}{r} \max \left(x_{1}+2 x_{2}+x_{3}\right) \\ \text { subject to } \quad x_{1}, x_{2}, x_{3} \geqslant 0 \\ x_{1}+x_{2}+2 x_{3} \leqslant 10 \\ 2 x_{1}+x_{2}+3 x_{3} \leqslant 15 \end{array}$$

  Suppose we now subtract $\Delta \in[0,10]$ from the right hand side of the last two constraints. Find the new optimal value.

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• Paper 2, Section I, H

  #Optimization

  2017

  Consider the following optimization problem

  $$\mathrm{P}: \quad \min f(x) \quad \text { subject to } \quad g(x)=b, x \in X .$$

  (a) Write down the Lagrangian for this problem. State the Lagrange sufficiency theorem.

  (b) Formulate the dual problem. State and prove the weak duality property.

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• Paper 4, Section II, H

  #Optimization

  2017

  (a) Let $G$ be a flow network with capacities $c_{i j}$ on the edges. Explain the maximum flow problem on this network defining all the notation you need.

  (b) Describe the Ford-Fulkerson algorithm for finding a maximum flow and state the max-flow min-cut theorem.

  (c) Apply the Ford-Fulkerson algorithm to find a maximum flow and a minimum cut of the following network:

  

  (d) Suppose that we add $\varepsilon>0$ to each capacity of a flow network. Is it true that the maximum flow will always increase by $\varepsilon$ ? Justify your answer.

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• Paper 3, Section II, H

  #Optimization

  2017

  (a) Explain what is meant by a two-person zero-sum game with payoff matrix $A=\left(a_{i j}: 1 \leqslant i \leqslant m, 1 \leqslant j \leqslant n\right)$ and define what is an optimal strategy (also known as a maximin strategy) for each player.

  (b) Suppose the payoff matrix $A$ is antisymmetric, i.e. $m=n$ and $a_{i j}=-a_{j i}$ for all $i, j$. What is the value of the game? Justify your answer.

  (c) Consider the following two-person zero-sum game. Let $n \geqslant 3$. Both players simultaneously call out one of the numbers $\{1, \ldots, n\}$. If the numbers differ by one, the player with the higher number wins $£ 1$ from the other player. If the players' choices differ by 2 or more, the player with the higher number pays $£ 2$ to the other player. In the event of a tie, no money changes hands.

  Write down the payoff matrix.

  For the case when $n=3$ find the value of the game and an optimal strategy for each player.

  Find the value of the game and an optimal strategy for each player for all $n$.

  [You may use results from the course provided you state them clearly.]

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• Paper 4, Section I, B

  #Quantum Mechanics

  2017

  (a) Give a physical interpretation of the wavefunction $\phi(x, t)=A e^{i k x} e^{-i E t / \hbar}$ (where $A, k$ and $E$ are real constants).

  (b) A particle of mass $m$ and energy $E>0$ is incident from the left on the potential step

  $$V(x)=\left\{\begin{array}{cl} 0 & \text { for }-\infty<x<a \\ V_{0} & \text { for } a<x<\infty \end{array}\right.$$

  with $V_{0}>0$.

  State the conditions satisfied by a stationary state at the point $x=a$.

  Compute the probability that the particle is reflected as a function of $E$, and compare your result with the classical case.

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• Paper 3, Section I, B

  #Quantum Mechanics

  2017

  A particle of mass $m$ is confined to a one-dimensional box $0 \leqslant x \leqslant a$. The potential $V(x)$ is zero inside the box and infinite outside.

  (a) Find the allowed energies of the particle and the normalised energy eigenstates.

  (b) At time $t=0$ the particle has wavefunction $\psi_{0}$ that is uniform in the left half of the box i.e. $\psi_{0}(x)=\sqrt{\frac{2}{a}}$ for $0<x<a / 2$ and $\psi_{0}(x)=0$ for $a / 2<x<a$. Find the probability that a measurement of energy at time $t=0$ will yield a value less than $5 \hbar^{2} \pi^{2} /\left(2 m a^{2}\right)$.

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• Paper 1, Section II, B

  #Quantum Mechanics

  2017

  Consider the time-independent Schrödinger equation in one dimension for a particle of mass $m$ with potential $V(x)$.

  (a) Show that if the potential is an even function then any non-degenerate stationary state has definite parity.

  (b) A particle of mass $m$ is subject to the potential $V(x)$ given by

  $$V(x)=-\lambda(\delta(x-a)+\delta(x+a))$$

  where $\lambda$ and $a$ are real positive constants and $\delta(x)$ is the Dirac delta function.

  Derive the conditions satisfied by the wavefunction $\psi(x)$ around the points $x=\pm a$.

  Show (using a graphical method or otherwise) that there is a bound state of even parity for any $\lambda>0$, and that there is an odd parity bound state only if $\lambda>\hbar^{2} /(2 m a)$. [Hint: You may assume without proof that the functions $x \tanh x$ and $x \operatorname{coth} x$ are monotonically increasing for $x>0$.]

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• Paper 3, Section II, B

  #Quantum Mechanics

  2017

  (a) Given the position and momentum operators $\hat{x}_{i}=x_{i}$ and $\hat{p}_{i}=-i \hbar \partial / \partial x_{i}$ (for $i=1,2,3)$ in three dimensions, define the angular momentum operators $\hat{L}_{i}$ and the total angular momentum $\hat{L}^{2}$.

  Show that $\hat{L}_{3}$ is Hermitian.

  (b) Derive the generalised uncertainty relation for the observables $\hat{L}_{3}$ and $\hat{x}_{1}$ in the form

  $$\Delta_{\psi} \hat{L}_{3} \Delta_{\psi} \hat{x}_{1} \geqslant M$$

  for any state $\psi$ and a suitable expression $M$ that you should determine. [Hint: It may be useful to consider the operator $\hat{L}_{3}+i \lambda \hat{x}_{1}$.]

  (c) Consider a particle with wavefunction

  $$\psi=K\left(x_{1}+x_{2}+2 x_{3}\right) e^{-\alpha r}$$

  where $r=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$ and $K$ and $\alpha$ are real positive constants.

  Show that $\psi$ is an eigenstate of total angular momentum $\hat{L}^{2}$ and find the corresponding angular momentum quantum number $l$. Find also the expectation value of a measurement of $\hat{L}_{3}$ on the state $\psi$.

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• Paper 2, Section II, B

  #Quantum Mechanics

  2017

  (a) The potential for the one-dimensional harmonic oscillator is $V(x)=\frac{1}{2} m \omega^{2} x^{2}$. By considering the associated time-independent Schrödinger equation for the wavefunction $\psi(x)$ with substitutions

  $$\xi=\left(\frac{m \omega}{\hbar}\right)^{1 / 2} x \quad \text { and } \quad \psi(x)=f(\xi) e^{-\xi^{2} / 2}$$

  show that the allowed energy levels are given by $E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega$ for $n=0,1,2, \ldots$ [You may assume without proof that $f$ must be a polynomial for $\psi$ to be normalisable.]

  (b) Consider a particle with charge $q$ and mass $m=1$ subject to the one-dimensional harmonic oscillator potential $U_{0}(x)=x^{2} / 2$. You may assume that the normalised ground state of this potential is

  $$\psi_{0}(x)=\left(\frac{1}{\pi \hbar}\right)^{1 / 4} e^{-x^{2} /(2 \hbar)}$$

  The particle is in the stationary state corresponding to $\psi_{0}(x)$ when at time $t=t_{0}$, an electric field of constant strength $E$ is turned on, adding an extra term $U_{1}(x)=-q E x$ to the harmonic potential.

  (i) Using the result of part (a) or otherwise, find the energy levels of the new potential.

  (ii) Show that the probability of finding the particle in the ground state immediately after $t_{0}$ is given by $e^{-q^{2} E^{2} /(2 \hbar)}$. [You may assume that $\int_{-\infty}^{\infty} e^{-x^{2}+2 A x} d x=$ $\sqrt{\pi} e^{A^{2}}$.]

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• Paper 1, Section I, H

  #Statistics

  2017

  (a) State and prove the Rao-Blackwell theorem.

  (b) Let $X_{1}, \ldots, X_{n}$ be an independent sample from $\operatorname{Poisson}(\lambda)$ with $\theta=e^{-\lambda}$ to be estimated. Show that $Y=1_{\{0\}}\left(X_{1}\right)$ is an unbiased estimator of $\theta$ and that $T=\sum_{i} X_{i}$ is a sufficient statistic.

  What is $\mathbb{E}[Y \mid T] ?$

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• Paper 2, Section I, 8H

  #Statistics

  2017

  (a) Define a $100 \gamma \%$ confidence interval for an unknown parameter $\theta$.

  (b) Let $X_{1}, \ldots, X_{n}$ be i.i.d. random variables with distribution $N(\mu, 1)$ with $\mu$ unknown. Find a $95 \%$ confidence interval for $\mu$.

  [You may use the fact that $\Phi(1.96) \simeq 0.975 .]$

  (c) Let $U_{1}, U_{2}$ be independent $U[\theta-1, \theta+1]$ with $\theta$ to be estimated. Find a $50 \%$ confidence interval for $\theta$.

  Suppose that we have two observations $u_{1}=10$ and $u_{2}=11.5$. What might be a better interval to report in this case?

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• Paper 4, Section II, H

  #Statistics

  2017

  (a) State and prove the Neyman-Pearson lemma.

  (b) Let $X$ be a real random variable with density $f(x)=(2 \theta x+1-\theta) 1_{[0,1]}(x)$ with $-1 \leqslant \theta \leqslant 1 .$

  Find a most powerful test of size $\alpha$ of $H_{0}: \theta=0$ versus $H_{1}: \theta=1$.

  Find a uniformly most powerful test of size $\alpha$ of $H_{0}: \theta=0$ versus $H_{1}: \theta>0$.

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• Paper 1, Section II, H

  #Statistics

  2017

  (a) Give the definitions of a sufficient and a minimal sufficient statistic $T$ for an unknown parameter $\theta$.

  Let $X_{1}, X_{2}, \ldots, X_{n}$ be an independent sample from the geometric distribution with success probability $1 / \theta$ and mean $\theta>1$, i.e. with probability mass function

  $$p(m)=\frac{1}{\theta}\left(1-\frac{1}{\theta}\right)^{m-1} \text { for } m=1,2, \ldots$$

  Find a minimal sufficient statistic for $\theta$. Is your statistic a biased estimator of $\theta ?$

  [You may use results from the course provided you state them clearly.]

  (b) Define the bias of an estimator. What does it mean for an estimator to be unbiased?

  Suppose that $Y$ has the truncated Poisson distribution with probability mass function

  $$p(y)=\left(e^{\theta}-1\right)^{-1} \cdot \frac{\theta^{y}}{y !} \quad \text { for } y=1,2, \ldots$$

  Show that the only unbiased estimator $T$ of $1-e^{-\theta}$ based on $Y$ is obtained by taking $T=0$ if $Y$ is odd and $T=2$ if $Y$ is even.

  Is this a useful estimator? Justify your answer.

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• Paper 3, Section II, $\mathbf{2 0 H}$

  #Statistics

  2017

  Consider the general linear model

  $$\boldsymbol{Y}=X \boldsymbol{\beta}+\varepsilon$$

  where $X$ is a known $n \times p$ matrix of full rank $p<n, \varepsilon \sim \mathcal{N}_{n}\left(0, \sigma^{2} I\right)$ with $\sigma^{2}$ known and $\boldsymbol{\beta} \in \mathbb{R}^{p}$ is an unknown vector.

  (a) State without proof the Gauss-Markov theorem.

  Find the maximum likelihood estimator $\widehat{\boldsymbol{\beta}}$ for $\boldsymbol{\beta}$. Is it unbiased?

  Let $\boldsymbol{\beta}^{*}$ be any unbiased estimator for $\boldsymbol{\beta}$ which is linear in $\left(Y_{i}\right)$. Show that

  $$\operatorname{var}\left(\boldsymbol{t}^{T} \widehat{\boldsymbol{\beta}}\right) \leqslant \operatorname{var}\left(\boldsymbol{t}^{T} \boldsymbol{\beta}^{*}\right)$$

  for all $\boldsymbol{t} \in \mathbb{R}^{p}$.

  (b) Suppose now that $p=1$ and that $\boldsymbol{\beta}$ and $\sigma^{2}$ are both unknown. Find the maximum likelihood estimator for $\sigma^{2}$. What is the joint distribution of $\widehat{\boldsymbol{\beta}}$ and $\widehat{\sigma}^{2}$ in this case? Justify your answer.

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• Paper 1, Section I, D

  #Variational Principles

  2017

  Derive the Euler-Lagrange equation for the function $u(x, y)$ that gives a stationary value of

  $$I[u]=\int_{\mathcal{D}} L\left(x, y, u, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}\right) d x d y$$

  where $\mathcal{D}$ is a bounded domain in the $(x, y)$-plane and $u$ is fixed on the boundary $\partial \mathcal{D}$.

  Find the equation satisfied by the function $u$ that gives a stationary value of

  $$I=\int_{\mathcal{D}}\left[\left(\frac{\partial u}{\partial x}\right)^{2}+\left(\frac{\partial u}{\partial y}\right)^{2}+k^{2} u^{2}\right] d x d y$$

  where $k$ is a constant and $u$ is prescribed on $\partial \mathcal{D}$.

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• Paper 3, Section $I$, D

  #Variational Principles

  2017

  (a) A Pringle crisp can be defined as the surface

  $$z=x y \quad \text { with } \quad x^{2}+y^{2} \leqslant 1$$

  Use the method of Lagrange multipliers to find the minimum and maximum values of $z$ on the boundary of the Pringle crisp and the $(x, y)$ positions where these occur.

  (b) A farmer wishes to construct a grain silo in the form of a hollow vertical cylinder of radius $r$ and height $h$ with a hollow hemispherical cap of radius $r$ on top of the cylinder. The walls of the cylinder cost $£ x$ per unit area to construct and the surface of the cap costs $£ 2 x$ per unit area to construct. Given that a total volume $V$ is desired for the silo, what values of $r$ and $h$ should be chosen to minimise the cost?

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• Paper 2, Section II, D

  #Variational Principles

  2017

  A proto-planet of mass $m$ in a uniform galactic dust cloud has kinetic and potential energies

  $$T=\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\phi}^{2}, \quad V=k m r^{2}$$

  where $k$ is constant. State Hamilton's principle and use it to determine the equations of motion for the proto-planet.

  Write down two conserved quantities of the motion and state why their existence illustrates Noether's theorem.

  Determine the Hamiltonian $H(\mathbf{p}, \mathbf{x})$ of this system, where $\mathbf{p}=\left(p_{r}, p_{\phi}\right), \mathbf{x}=(r, \phi)$ and $\left(p_{r}, p_{\phi}\right)$ are the conjugate momenta corresponding to $(r, \phi)$.

  Write down Hamilton's equations for this system and use them to show that

  $$m \ddot{r}=-V_{\text {eff }}^{\prime}(r), \quad \text { where } \quad V_{\text {eff }}(r)=m\left(\frac{h^{2}}{2 m^{2} r^{2}}+k r^{2}\right)$$

  and $h$ is a constant. With the aid of a diagram, explain why there is a stable circular orbit.

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• Paper 4, Section II, $16 \mathrm{D}$

  #Variational Principles

  2017

  Consider the functional

  $$F[y]=\int_{\alpha}^{\beta} f\left(y, y^{\prime}, x\right) d x$$

  of a function $y(x)$ defined for $x \in[\alpha, \beta]$, with $y$ having fixed values at $x=\alpha$ and $x=\beta$.

  By considering $F[y+\epsilon \xi]$, where $\xi(x)$ is an arbitrary function with $\xi(\alpha)=\xi(\beta)=0$ and $\epsilon \ll 1$, determine that the second variation of $F$ is

  $$\delta^{2} F[y, \xi]=\int_{\alpha}^{\beta}\left\{\xi^{2}\left[\frac{\partial^{2} f}{\partial y^{2}}-\frac{d}{d x}\left(\frac{\partial^{2} f}{\partial y \partial y^{\prime}}\right)\right]+\xi^{\prime 2} \frac{\partial^{2} f}{\partial y^{\prime 2}}\right\} d x$$

  The surface area of an axisymmetric soap film joining two parallel, co-axial, circular rings of radius a distance $2 L$ apart can be expressed by the functional

  $$F[y]=\int_{-L}^{L} 2 \pi y \sqrt{1+y^{\prime 2}} d x$$

  where $x$ is distance in the axial direction and $y$ is radial distance from the axis. Show that the surface area is stationary when

  $$y=E \cosh \frac{x}{E},$$

  where $E$ is a constant that need not be determined, and that the stationary area is a local minimum if

  $$\int_{-L / E}^{L / E}\left(\xi^{\prime 2}-\xi^{2}\right) \operatorname{sech}^{2} z d z>0$$

  for all functions $\xi(z)$ that vanish at $z=\pm L / E$, where $z=x / E$.

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• Paper 2, Section II, I

  #Algebraic Geometry

  2017

  Let $k$ be an algebraically closed field of any characteristic.

  (a) Define what it means for a variety $X$ to be non-singular at a point $P \in X$.

  (b) Let $X \subseteq \mathbb{P}^{n}$ be a hypersurface $Z(f)$ for $f \in k\left[x_{0}, \ldots, x_{n}\right]$ an irreducible homogeneous polynomial. Show that the set of singular points of $X$ is $Z(I)$, where $I \subseteq$ $k\left[x_{0}, \ldots, x_{n}\right]$ is the ideal generated by $\partial f / \partial x_{0}, \ldots, \partial f / \partial x_{n} .$

  (c) Consider the projective plane curve corresponding to the affine curve in $\mathbb{A}^{2}$ given by the equation

  $$x^{4}+x^{2} y^{2}+y^{2}+1=0 .$$

  Find the singular points of this projective curve if char $k \neq 2$. What goes wrong if char $k=2$ ?

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• Paper 3, Section II, I

  #Algebraic Geometry

  2017

  (a) Define what it means to give a rational map between algebraic varieties. Define a birational map.

  (b) Let

  $$X=Z\left(y^{2}-x^{2}(x-1)\right) \subseteq \mathbb{A}^{2}$$

  Define a birational map from $X$ to $\mathbb{A}^{1}$. [Hint: Consider lines through the origin.]

  (c) Let $Y \subseteq \mathbb{A}^{3}$ be the surface given by the equation

  $$x_{1}^{2} x_{2}+x_{2}^{2} x_{3}+x_{3}^{2} x_{1}=0 .$$

  Consider the blow-up $X \subseteq \mathbb{A}^{3} \times \mathbb{P}^{2}$ of $\mathbb{A}^{3}$ at the origin, i.e. the subvariety of $\mathbb{A}^{3} \times \mathbb{P}^{2}$ defined by the equations $x_{i} y_{j}=x_{j} y_{i}$ for $1 \leqslant i<j \leqslant 3$, with $y_{1}, y_{2}, y_{3}$ coordinates on $\mathbb{P}^{2}$. Let $\varphi: X \rightarrow \mathbb{A}^{3}$ be the projection and $E=\varphi^{-1}(0)$. Recall that the proper transform $\tilde{Y}$ of $Y$ is the closure of $\varphi^{-1}(Y) \backslash E$ in $X$. Give equations for $\tilde{Y}$, and describe the fibres of the morphism $\left.\varphi\right|_{\widetilde{Y}}: \widetilde{Y} \rightarrow Y$.

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• Paper 4, Section II, I

  #Algebraic Geometry

  2017

  (a) Let $X$ and $Y$ be non-singular projective curves over a field $k$ and let $\varphi: X \rightarrow Y$ be a non-constant morphism. Define the ramification degree $e_{P}$ of $\varphi$ at a point $P \in X$.

  (b) Suppose char $k \neq 2$. Let $X=Z(f)$ be the plane cubic with $f=x_{0} x_{2}^{2}-x_{1}^{3}+x_{0}^{2} x_{1}$, and let $Y=\mathbb{P}^{1}$. Explain how the projection

  $$\left(x_{0}: x_{1}: x_{2}\right) \mapsto\left(x_{0}: x_{1}\right)$$

  defines a morphism $\varphi: X \rightarrow Y$. Determine the degree of $\varphi$ and the ramification degrees $e_{P}$ for all $P \in X$.

  (c) Let $X$ be a non-singular projective curve and let $P \in X$. Show that there is a non-constant rational function on $X$ which is regular on $X \backslash\{P\}$.

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• Paper 1, Section II, I

  #Algebraic Geometry

  2017

  Let $k$ be an algebraically closed field.

  (a) Let $X$ and $Y$ be varieties defined over $k$. Given a function $f: X \rightarrow Y$, define what it means for $f$ to be a morphism of varieties.

  (b) If $X$ is an affine variety, show that the coordinate ring $A(X)$ coincides with the ring of regular functions on $X$. [Hint: You may assume a form of the Hilbert Nullstellensatz.]

  (c) Now suppose $X$ and $Y$ are affine varieties. Show that if $X$ and $Y$ are isomorphic, then there is an isomorphism of $k$-algebras $A(X) \cong A(Y)$.

  (d) Show that $Z\left(x^{2}-y^{3}\right) \subseteq \mathbb{A}^{2}$ is not isomorphic to $\mathbb{A}^{1}$.

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• Paper 3, Section II, I

  #Algebraic Topology

  2017

  The $n$-torus is the product of $n$ circles:

  $$T^{n}=\underbrace{S^{1} \times \ldots \times S^{1}}_{n \text { times }} .$$

  For all $n \geqslant 1$ and $0 \leqslant k \leqslant n$, compute $H_{k}\left(T^{n}\right)$.

  [You may assume that relevant spaces are triangulable, but you should state carefully any version of any theorem that you use.]

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• Paper 2, Section II, I

  #Algebraic Topology

  2017

  (a) (i) Define the push-out of the following diagram of groups.

  

  When is a push-out a free product with amalgamation?

  (ii) State the Seifert-van Kampen theorem.

  (b) Let $X=\mathbb{R} P^{2} \vee S^{1}$ (recalling that $\mathbb{R} P^{2}$ is the real projective plane), and let $x \in X$.

  (i) Compute the fundamental group $\pi_{1}(X, x)$ of the space $X$.

  (ii) Show that there is a surjective homomorphism $\phi: \pi_{1}(X, x) \rightarrow S_{3}$, where $S_{3}$ is the symmetric group on three elements.

  (c) Let $\widehat{X} \rightarrow X$ be the covering space corresponding to the kernel of $\phi$.

  (i) Draw $\widehat{X}$ and justify your answer carefully.

  (ii) Does $\widehat{X}$ retract to a graph? Justify your answer briefly.

  (iii) Does $\widehat{X}$ deformation retract to a graph? Justify your answer briefly.

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• Paper 1, Section II, I

  #Algebraic Topology

  2017

  Let $X$ be a topological space and let $x_{0}$ and $x_{1}$ be points of $X$.

  (a) Explain how a path $u:[0,1] \rightarrow X$ from $x_{0}$ to $x_{1}$ defines a map $u_{\sharp}: \pi_{1}\left(X, x_{0}\right) \rightarrow$ $\pi_{1}\left(X, x_{1}\right)$.

  (b) Prove that $u_{\sharp}$ is an isomorphism of groups.

  (c) Let $\alpha, \beta:\left(S^{1}, 1\right) \rightarrow\left(X, x_{0}\right)$ be based loops in $X$. Suppose that $\alpha, \beta$ are homotopic as unbased maps, i.e. the homotopy is not assumed to respect basepoints. Show that the corresponding elements of $\pi_{1}\left(X, x_{0}\right)$ are conjugate.

  (d) Take $X$ to be the 2-torus $S^{1} \times S^{1}$. If $\alpha, \beta$ are homotopic as unbased loops as in part (c), then exhibit a based homotopy between them. Interpret this fact algebraically.

  (e) Exhibit a pair of elements in the fundamental group of $S^{1} \vee S^{1}$ which are homotopic as unbased loops but not as based loops. Justify your answer.

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• Paper 4, Section II, I

  #Algebraic Topology

  2017

  Recall that $\mathbb{R} P^{n}$ is real projective $n$-space, the quotient of $S^{n}$ obtained by identifying antipodal points. Consider the standard embedding of $S^{n}$ as the unit sphere in $\mathbb{R}^{n+1}$.

  (a) For $n$ odd, show that there exists a continuous map $f: S^{n} \rightarrow S^{n}$ such that $f(x)$ is orthogonal to $x$, for all $x \in S^{n}$.

  (b) Exhibit a triangulation of $\mathbb{R} P^{n}$.

  (c) Describe the map $H_{n}\left(S^{n}\right) \rightarrow H_{n}\left(S^{n}\right)$ induced by the antipodal map, justifying your answer.

  (d) Show that, for $n$ even, there is no continuous map $f: S^{n} \rightarrow S^{n}$ such that $f(x)$ is orthogonal to $x$ for all $x \in S^{n}$.

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• Paper 3, Section II, F

  #Analysis of Functions

  2017

  Denote by $C_{0}\left(\mathbb{R}^{n}\right)$ the space of continuous complex-valued functions on $\mathbb{R}^{n}$ converging to zero at infinity. Denote by $\mathcal{F} f(\xi)=\int_{\mathbb{R}^{n}} e^{-2 i \pi x \cdot \xi} f(x) d x$ the Fourier transform of $f \in L^{1}\left(\mathbb{R}^{n}\right)$.

  (i) Prove that the image of $L^{1}\left(\mathbb{R}^{n}\right)$ under $\mathcal{F}$ is included and dense in $C_{0}\left(\mathbb{R}^{n}\right)$, and that $\mathcal{F}: L^{1}\left(\mathbb{R}^{n}\right) \rightarrow C_{0}\left(\mathbb{R}^{n}\right)$ is injective. [Fourier inversion can be used without proof when properly stated.]

  (ii) Calculate the Fourier transform of $\chi_{[a, b]}$, the characteristic function of $[a, b] \subset \mathbb{R}$.

  (iii) Prove that $g_{n}:=\chi_{[-n, n]} * \chi_{[-1,1]}$ belongs to $C_{0}(\mathbb{R})$ and is the Fourier transform of a function $h_{n} \in L^{1}(\mathbb{R})$, which you should determine.

  (iv) Using the functions $h_{n}, g_{n}$ and the open mapping theorem, deduce that the Fourier transform is not surjective from $L^{1}(\mathbb{R})$ to $C_{0}(\mathbb{R})$.

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• Paper 4, Section II, $22 F$

  #Analysis of Functions

  2017

  Consider $\mathbb{R}^{n}$ with the Lebesgue measure. Denote by $\mathcal{F} f(\xi)=\int_{\mathbb{R}^{n}} e^{-2 i \pi x \cdot \xi} f(x) d x$ the Fourier transform of $f \in L^{1}\left(\mathbb{R}^{n}\right)$ and by $\hat{f}$ the Fourier-Plancherel transform of $f \in L^{2}\left(\mathbb{R}^{n}\right)$. Let $\chi_{R}(\xi):=\left(1-\frac{|\xi|}{R}\right) \chi_{|\xi| \leqslant R}$ for $R>0$ and define for $s \in \mathbb{R}_{+}$

  $$H^{s}\left(\mathbb{R}^{n}\right):=\left\{f \in L^{2}\left(\mathbb{R}^{n}\right) \mid\left(1+|\cdot|^{2}\right)^{s / 2} \hat{f}(\cdot) \in L^{2}\left(\mathbb{R}^{n}\right)\right\}$$

  (i) Prove that $H^{s}\left(\mathbb{R}^{n}\right)$ is a vector subspace of $L^{2}\left(\mathbb{R}^{n}\right)$, and is a Hilbert space for the inner product $\langle f, g\rangle:=\int_{\mathbb{R}^{n}}\left(1+|\xi|^{2}\right)^{s} \hat{f}(\xi) \overline{\hat{g}(\xi)} d \xi$, where $\bar{z}$ denotes the complex conjugate of $z \in \mathbb{C}$.

  (ii) Construct a function $f \in H^{s}(\mathbb{R}), s \in(0,1 / 2)$, that is not almost everywhere equal to a continuous function.

  (iii) For $f \in L^{1}\left(\mathbb{R}^{n}\right)$, prove that $F_{R}: x \mapsto \int_{\mathbb{R}^{n}} \mathcal{F} f(\xi) \chi_{R}(\xi) e^{2 i \pi x \cdot \xi} d \xi$ is a well-defined function and that $F_{R} \in L^{1}\left(\mathbb{R}^{n}\right)$ converges to $f$ in $L^{1}\left(\mathbb{R}^{n}\right)$ as $R \rightarrow+\infty$.

  [Hint: Prove that $F_{R}=K_{R} * f$ where $K_{R}$ is an approximation of the unit as $R \rightarrow+\infty .]$

  (iv) Deduce that if $f \in L^{1}\left(\mathbb{R}^{n}\right)$ and $\left(1+|\cdot|^{2}\right)^{s / 2} \mathcal{F} f(\cdot) \in L^{2}\left(\mathbb{R}^{n}\right)$ then $f \in H^{s}\left(\mathbb{R}^{n}\right)$.

  [Hint: Prove that: (1) there is a sequence $R_{k} \rightarrow+\infty$ such that $K_{R_{k}} * f$ converges to $f$ almost everywhere; (2) $K_{R} * f$ is uniformly bounded in $L^{2}\left(\mathbb{R}^{n}\right)$ as $R \rightarrow+\infty$.]

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• Paper 1, Section II, $22 F$

  #Analysis of Functions

  2017

  Consider a sequence $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$ of measurable functions converging pointwise to a function $f: \mathbb{R} \rightarrow \mathbb{R}$. The Lebesgue measure is denoted by $\lambda$.

  (a) Consider a Borel set $A \subset \mathbb{R}$ with finite Lebesgue measure $\lambda(A)<+\infty$. Define for $k, n \geqslant 1$ the sets

  $$E_{n}^{(k)}:=\bigcap_{m \geqslant n}\left\{x \in A|| f_{m}(x)-f(x) \mid \leqslant \frac{1}{k}\right\}$$

  Prove that for any $k, n \geqslant 1$, one has $E_{n}^{(k)} \subset E_{n+1}^{(k)}$ and $E_{n}^{(k+1)} \subset E_{n}^{(k)}$. Prove that for any $k \geqslant 1, A=\cup_{n \geqslant 1} E_{n}^{(k)}$.

  (b) Consider a Borel set $A \subset \mathbb{R}$ with finite Lebesgue measure $\lambda(A)<+\infty$. Prove that for any $\varepsilon>0$, there is a Borel set $A_{\varepsilon} \subset A$ for which $\lambda\left(A \backslash A_{\varepsilon}\right) \leqslant \varepsilon$ and such that $f_{n}$ converges to $f$ uniformly on $A_{\varepsilon}$ as $n \rightarrow+\infty$. Is the latter still true when $\lambda(A)=+\infty$ ?

  (c) Assume additionally that $f_{n} \in L^{p}(\mathbb{R})$ for some $p \in(1,+\infty]$, and there exists an $M \geqslant 0$ for which $\left\|f_{n}\right\|_{L^{p}(\mathbb{R})} \leqslant M$ for all $n \geqslant 1$. Prove that $f \in L^{p}(\mathbb{R})$.

  (d) Let $f_{n}$ and $f$ be as in part (c). Consider a Borel set $A \subset \mathbb{R}$ with finite Lebesgue measure $\lambda(A)<+\infty$. Prove that $f_{n}, f$ are integrable on $A$ and $\int_{A} f_{n} d \lambda \rightarrow \int_{A} f d \lambda$ as $n \rightarrow \infty$. Deduce that $f_{n}$ converges weakly to $f$ in $L^{p}(\mathbb{R})$ when $p<+\infty$. Does the convergence have to be strong?

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• Paper 1, Section II, C

  #Applications of Quantum Mechanics

  2017

  A one-dimensional lattice has $N$ sites with lattice spacing $a$. In the tight-binding approximation, the Hamiltonian describing a single electron is given by

  $$H=E_{0} \sum_{n}|n\rangle\langle n|-J \sum_{n}(|n\rangle\langle n+1|+| n+1\rangle\langle n|)$$

  where $|n\rangle$ is the normalised state of the electron localised on the $n^{\text {th }}$ lattice site. Using periodic boundary conditions $|N+1\rangle \equiv|1\rangle$, solve for the spectrum of this Hamiltonian to derive the dispersion relation

  $$E(k)=E_{0}-2 J \cos (k a)$$

  Define the Brillouin zone. Determine the number of states in the Brillouin zone.

  Calculate the velocity $v$ and effective mass $m^{\star}$ of the particle. For which values of $k$ is the effective mass negative?

  In the semi-classical approximation, derive an expression for the time-dependence of the position of the electron in a constant electric field.

  Describe how the concepts of metals and insulators arise in the model above.

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• Paper 2, Section II, C

  #Applications of Quantum Mechanics

  2017

  Give an account of the variational method for establishing an upper bound on the ground-state energy of a Hamiltonian $H$ with a discrete spectrum $H|n\rangle=E_{n}|n\rangle$, where $E_{n} \leqslant E_{n+1}, n=0,1,2 \ldots$

  A particle of mass $m$ moves in the three-dimensional potential

  $$V(r)=-\frac{A e^{-\mu r}}{r}$$

  where $A, \mu>0$ are constants and $r$ is the distance to the origin. Using the normalised variational wavefunction

  $$\psi(r)=\sqrt{\frac{\alpha^{3}}{\pi}} e^{-\alpha r}$$

  show that the expected energy is given by

  $$E(\alpha)=\frac{\hbar^{2} \alpha^{2}}{2 m}-\frac{4 A \alpha^{3}}{(\mu+2 \alpha)^{2}}$$

  Explain why there is necessarily a bound state when $\mu<A m / \hbar^{2}$. What can you say about the existence of a bound state when $\mu \geqslant A m / \hbar^{2}$ ?

  [Hint: The Laplacian in spherical polar coordinates is

  $$\left.\nabla^{2}=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2}}{\partial \phi^{2}}\right]$$

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• Paper 3, Section II, C

  #Applications of Quantum Mechanics

  2017

  A particle of mass $m$ and charge $q$ moving in a uniform magnetic field $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}=$ $(0,0, B)$ is described by the Hamiltonian

  $$H=\frac{1}{2 m}(\mathbf{p}-q \mathbf{A})^{2}$$

  where $\mathbf{p}$ is the canonical momentum, which obeys $\left[x_{i}, p_{j}\right]=i \hbar \delta_{i j}$. The mechanical momentum is defined as $\boldsymbol{\pi}=\mathbf{p}-q \mathbf{A}$. Show that

  $$\left[\pi_{x}, \pi_{y}\right]=i q \hbar B$$

  Define

  $$a=\frac{1}{\sqrt{2 q \hbar B}}\left(\pi_{x}+i \pi_{y}\right) \quad \text { and } \quad a^{\dagger}=\frac{1}{\sqrt{2 q \hbar B}}\left(\pi_{x}-i \pi_{y}\right) \text {. }$$

  Derive the commutation relation obeyed by $a$ and $a^{\dagger}$. Write the Hamiltonian in terms of $a$ and $a^{\dagger}$ and hence solve for the spectrum.

  In symmetric gauge, states in the lowest Landau level with $k_{z}=0$ have wavefunctions

  $$\psi(x, y)=(x+i y)^{M} e^{-q B r^{2} / 4 \hbar}$$

  where $r^{2}=x^{2}+y^{2}$ and $M$ is a positive integer. By considering the profiles of these wavefunctions, estimate how many lowest Landau level states can fit in a disc of radius $R$.

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• Paper 4, Section II, C

  #Applications of Quantum Mechanics

  2017

  (a) In one dimension, a particle of mass $m$ is scattered by a potential $V(x)$ where $V(x) \rightarrow 0$ as $|x| \rightarrow \infty$. For wavenumber $k>0$, the incoming $(\mathcal{I})$ and outgoing $(\mathcal{O})$ asymptotic plane wave states with positive $(+)$ and negative $(-)$ parity are given by

  $$\begin{array}{rr} \mathcal{I}_{+}(x)=e^{-i k|x|}, & \mathcal{I}_{-}(x)=\operatorname{sign}(x) e^{-i k|x|} \\ \mathcal{O}_{+}(x)=e^{+i k|x|}, & \mathcal{O}_{-}(x)=-\operatorname{sign}(x) e^{+i k|x|} \end{array}$$

  (i) Explain how this basis may be used to define the $S$-matrix,

  $$\mathcal{S}^{P}=\left(\begin{array}{cc} S_{++} & S_{+-} \\ S_{-+} & S_{--} \end{array}\right)$$

  (ii) For what choice of potential would you expect $S_{+-}=S_{-+}=0$ ? Why?

  (b) The potential $V(x)$ is given by

  $$V(x)=V_{0}[\delta(x-a)+\delta(x+a)]$$

  with $V_{0}$ a constant.

  (i) Show that

  $$S_{--}(k)=e^{-2 i k a}\left[\frac{\left(2 k-i U_{0}\right) e^{i k a}+i U_{0} e^{-i k a}}{\left(2 k+i U_{0}\right) e^{-i k a}-i U_{0} e^{i k a}}\right]$$

  where $U_{0}=2 m V_{0} / \hbar^{2}$. Verify that $\left|S_{--}\right|^{2}=1$. Explain the physical meaning of this result.

  (ii) For $V_{0}<0$, by considering the poles or zeros of $S_{--}(k)$, show that there exists one bound state of negative parity if $a U_{0}<-1$.

  (iii) For $V_{0}>0$ and $a U_{0} \gg 1$, show that $S_{--}(k)$ has a pole at

  $$k a=\pi+\alpha-i \gamma$$

  where $\alpha$ and $\gamma$ are real and

  $$\alpha=-\frac{\pi}{a U_{0}}+O\left(\frac{1}{\left(a U_{0}\right)^{2}}\right) \quad \text { and } \quad \gamma=\left(\frac{\pi}{a U_{0}}\right)^{2}+O\left(\frac{1}{\left(a U_{0}\right)^{3}}\right)$$

  Explain the physical significance of this result.

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• Paper 2, Section II, K

  #Applied Probability

  2017

  (a) Give the definition of a Poisson process on $\mathbb{R}_{+}$. Let $X$ be a Poisson process on $\mathbb{R}_{+}$. Show that conditional on $\left\{X_{t}=n\right\}$, the jump times $J_{1}, \ldots, J_{n}$ have joint density function

  $$f\left(t_{1}, \ldots, t_{n}\right)=\frac{n !}{t^{n}} \mathbf{1}\left(0 \leqslant t_{1} \leqslant \ldots \leqslant t_{n} \leqslant t\right)$$

  where $\boldsymbol{I}(A)$ is the indicator of the set $A$.

  (b) Let $N$ be a Poisson process on $\mathbb{R}_{+}$with intensity $\lambda$ and jump times $\left\{J_{k}\right\}$. If $g: \mathbb{R}_{+} \rightarrow \mathbb{R}$ is a real function, we define for all $t>0$

  $$\mathcal{R}(g)[0, t]=\left\{g\left(J_{k}\right): k \in \mathbb{N}, J_{k} \leqslant t\right\}$$

  Show that for all $t>0$ the following is true

  $$\mathbb{P}(0 \in \mathcal{R}(g)[0, t])=1-\exp \left(-\lambda \int_{0}^{t} \mathbf{I}(g(s)=0) d s\right)$$

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• Paper 3, Section II, K

  #Applied Probability

  2017

  (a) Define the Moran model and Kingman's $n$-coalescent. Define Kingman's infinite coalescent.

  Show that Kingman's infinite coalescent comes down from infinity. In other words, with probability one, the number of blocks of $\Pi_{t}$ is finite at any time $t>0$.

  (b) Give the definition of a renewal process.

  Let $\left(X_{i}\right)$ denote the sequence of inter-arrival times of the renewal process $N$. Suppose that $\mathbb{E}\left[X_{1}\right]>0$.

  Prove that $\mathbb{P}(N(t) \rightarrow \infty$ as $t \rightarrow \infty)=1$.

  Prove that $\mathbb{E}\left[e^{\theta N(t)}\right]<\infty$ for some strictly positive $\theta$.

  [Hint: Consider the renewal process with inter-arrival times $X_{k}^{\prime}=\varepsilon \mathbf{1}\left(X_{k} \geqslant \varepsilon\right)$ for some suitable $\varepsilon>0$.]

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• Paper 4, Section II, $26 K$

  #Applied Probability

  2017

  (a) Give the definition of an $M / M / 1$ queue. Prove that if $\lambda$ is the arrival rate and $\mu$ the service rate and $\lambda<\mu$, then the length of the queue is a positive recurrent Markov chain. What is the equilibrium distribution?

  If the queue is in equilibrium and a customer arrives at some time $t$, what is the distribution of the waiting time (time spent waiting in the queue plus service time)?

  (b) We now modify the above queue: on completion of service a customer leaves with probability $\delta$, or goes to the back of the queue with probability $1-\delta$. Find the distribution of the total time a customer spends being served.

  Hence show that equilibrium is possible if $\lambda<\delta \mu$ and find the stationary distribution.

  Show that, in equilibrium, the departure process is Poisson.

  [You may use relevant theorems provided you state them clearly.]

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• Paper 1, Section II, $27 \mathrm{~K}$

  #Applied Probability

  2017

  (a) Define a continuous time Markov chain $X$ with infinitesimal generator $Q$ and jump chain $Y$.

  (b) Let $i$ be a transient state of a continuous-time Markov chain $X$ with $X(0)=i$. Show that the time spent in state $i$ has an exponential distribution and explicitly state its parameter.

  [You may use the fact that if $S \sim \operatorname{Exp}(\lambda)$, then $\mathbb{E}\left[e^{\theta S}\right]=\lambda /(\lambda-\theta)$ for $\theta<\lambda$.]

  (c) Let $X$ be an asymmetric random walk in continuous time on the non-negative integers with reflection at 0 , so that

  $$q_{i, j}= \begin{cases}\lambda & \text { if } \quad j=i+1, i \geqslant 0 \\ \mu & \text { if } \quad j=i-1, i \geqslant 1\end{cases}$$

  Suppose that $X(0)=0$ and $\lambda>\mu$. Show that for all $r \geqslant 1$, the total time $T_{r}$ spent in state $r$ is exponentially distributed with parameter $\lambda-\mu$.

  Assume now that $X(0)$ has some general distribution with probability generating function $G$. Find the expected amount of time spent at 0 in terms of $G$.

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• Paper 2, Section II, E

  #Asymptotic Methods

  2017

  Consider the function

  $$f_{\nu}(x) \equiv \frac{1}{2 \pi} \int_{C} \exp [-i x \sin z+i \nu z] d z$$

  where the contour $C$ is the boundary of the half-strip $\{z:-\pi<\operatorname{Re} z<\pi$ and $\operatorname{Im} z>0\}$, taken anti-clockwise.

  Use integration by parts and the method of stationary phase to:

  (i) Obtain the leading term for $f_{\nu}(x)$ coming from the vertical lines $z=\pm \pi+i y(0<$ $y<+\infty)$ for large $x>0$.

  (ii) Show that the leading term in the asymptotic expansion of the function $f_{\nu}(x)$ for large positive $x$ is

  $$\sqrt{\frac{2}{\pi x}} \cos \left(x-\frac{1}{2} \nu \pi-\frac{\pi}{4}\right)$$

  and obtain an estimate for the remainder as $O\left(x^{-a}\right)$ for some $a$ to be determined.

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• Paper 3, Section II, E

  #Asymptotic Methods

  2017

  Consider the integral representation for the modified Bessel function

  $$I_{0}(x)=\frac{1}{2 \pi i} \oint_{C} t^{-1} \exp \left[\frac{i x}{2}\left(t-\frac{1}{t}\right)\right] d t$$

  where $C$ is a simple closed contour containing the origin, taken anti-clockwise.

  Use the method of steepest descent to determine the full asymptotic expansion of $I_{0}(x)$ for large real positive $x .$

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• Paper 4, Section II, E

  #Asymptotic Methods

  2017

  Consider solutions to the equation

  $$\frac{d^{2} y}{d x^{2}}=\left(\frac{1}{4}+\frac{\mu^{2}-\frac{1}{4}}{x^{2}}\right) y$$

  of the form

  $$y(x)=\exp \left[S_{0}(x)+S_{1}(x)+S_{2}(x)+\ldots\right]$$

  with the assumption that, for large positive $x$, the function $S_{j}(x)$ is small compared to $S_{j-1}(x)$ for all $j=1,2 \ldots$

  Obtain equations for the $S_{j}(x), j=0,1,2 \ldots$, which are formally equivalent to ( $)$. Solve explicitly for $S_{0}$ and $S_{1}$. Show that it is consistent to assume that $S_{j}(x)=c_{j} x^{-(j-1)}$ for some constants $c_{j}$. Give a recursion relation for the $c_{j}$.

  Deduce that there exist two linearly independent solutions to $(\star)$ with asymptotic expansions as $x \rightarrow+\infty$ of the form

  $$y_{\pm}(x) \sim e^{\pm x / 2}\left(1+\sum_{j=1}^{\infty} A_{j}^{\pm} x^{-j}\right)$$

  Determine a recursion relation for the $A_{j}^{\pm}$. Compute $A_{1}^{\pm}$and $A_{2}^{\pm}$.

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• Paper 1, Section I, $4 \mathrm{H}$

  #Automata and Formal Languages

  2017

  (a) Prove that every regular language is also a context-free language (CFL).

  (b) Show that, for any fixed $n>0$, the set of regular expressions over the alphabet $\left\{a_{1}, \ldots, a_{n}\right\}$ is a CFL, but not a regular language.

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• Paper 2, Section I, $4 \mathrm{H}$

  #Automata and Formal Languages

  2017

  (a) Give explicit examples, with justification, of a language over some finite alphabet $\Sigma$ which is:

  (i) context-free, but not regular;

  (ii) recursive, but not context-free.

  (b) Give explicit examples, with justification, of a subset of $\mathbb{N}$ which is:

  (i) recursively enumerable, but not recursive;

  (ii) neither recursively enumerable, nor having recursively enumerable complement in $\mathbb{N}$.

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• Paper 3, Section I, 4H

  #Automata and Formal Languages

  2017

  (a) Define what it means for a context-free grammar (CFG) to be in Chomsky normal form (CNF). Give an example, with justification, of a context-free language (CFL) which is not defined by any CFG in CNF.

  (b) Show that the intersection of two CFLs need not be a CFL.

  (c) Let $L$ be a CFL over an alphabet $\Sigma$. Show that $\Sigma^{*} \backslash L$ need not be a CFL.

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• Paper 4, Section I, $4 \mathrm{H}$

  #Automata and Formal Languages

  2017

  (a) Describe the process for converting a deterministic finite-state automaton $D$ into a regular expression $R$ defining the same language, $\mathcal{L}(D)=\mathcal{L}(R)$. [You need only outline the steps, without proof, but you should clearly define all terminology you introduce.]

  (b) Consider the language $L$ over the alphabet $\{0,1\}$ defined via

  $$L:=\left\{w 01^{n} \mid w \in\{0,1\}^{*}, n \in \mathbb{K}\right\} \cup\{1\}^{*} .$$

  Show that $L$ satisfies the pumping lemma for regular languages but is not a regular language itself.

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• Paper 1, Section II, $11 H$

  #Automata and Formal Languages

  2017

  (a) Give an encoding to integers of all deterministic finite-state automata (DFAs). [Here the alphabet of each such DFA is always taken from the set $\{0,1, \ldots\}$, and the states for each such DFA are always taken from the set $\left.\left\{q_{0}, q_{1}, \ldots\right\} .\right]$

  (b) Show that the set of codes for which the corresponding DFA $D_{n}$ accepts a finite language is recursive. Moreover, if the language $\mathcal{L}\left(D_{n}\right)$ is finite, show that we can compute its size $\left|\mathcal{L}\left(D_{n}\right)\right|$ from $n$.

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• Paper 3, Section II, $11 \mathrm{H}$ Automata and formal languages

  #Automata and Formal Languages

  2017

  (a) Given $A, B \subseteq \mathbb{N}$, define a many-one reduction of $A$ to $B$. Show that if $B$ is recursively enumerable (r.e.) and $A \leqslant_{m} B$ then $A$ is also recursively enumerable.

  (b) State the $s-m-n$ theorem, and use it to prove that a set $X \subseteq \mathbb{N}$ is r.e. if and only if $X \leqslant_{m} \mathbb{K}$.

  (c) Consider the sets of integers $P, Q \subseteq \mathbb{N}$ defined via

  $$\begin{aligned} &P:=\left\{n \in \mathbb{N} \mid n \text { codes a program and } W_{n} \text { is finite }\right\} \\ &Q:=\left\{n \in \mathbb{N} \mid n \text { codes a program and } W_{n} \text { is recursive }\right\} \end{aligned}$$

  Show that $P \leqslant \leqslant_{m} Q$.

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• Paper 1, Section I, E

  #Classical Dynamics

  2017

  Consider a Lagrangian system with Lagrangian $L\left(x_{A}, \dot{x}_{A}, t\right)$, where $A=1, \ldots, 3 N$, and constraints

  $$f_{\alpha}\left(x_{A}, t\right)=0, \quad \alpha=1, \ldots, 3 N-n .$$

  Use the method of Lagrange multipliers to show that this is equivalent to a system with Lagrangian $\mathcal{L}\left(q_{i}, \dot{q}_{i}, t\right) \equiv L\left(x_{A}\left(q_{i}, t\right), \dot{x}_{A}\left(q_{i}, \dot{q}_{i}, t\right), t\right)$, where $i=1, \ldots, n$, and $q_{i}$ are coordinates on the surface of constraints.

  Consider a bead of unit mass in $\mathbb{R}^{2}$ constrained to move (with no potential) on a wire given by an equation $y=f(x)$, where $(x, y)$ are Cartesian coordinates. Show that the Euler-Lagrange equations take the form

  $$\frac{d}{d t} \frac{\partial \mathcal{L}}{\partial \dot{x}}=\frac{\partial \mathcal{L}}{\partial x}$$

  for some $\mathcal{L}=\mathcal{L}(x, \dot{x})$ which should be specified. Find one first integral of the EulerLagrange equations, and thus show that

  $$t=F(x)$$

  where $F(x)$ should be given in the form of an integral.

  [Hint: You may assume that the Euler-Lagrange equations hold in all coordinate systems.]

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• Paper 2, Section I, E

  #Classical Dynamics

  2017

  Derive the Lagrange equations from the principle of stationary action

  $$S[q]=\int_{t_{0}}^{t_{1}} \mathcal{L}\left(q_{i}(t), \dot{q}_{i}(t), t\right) d t, \quad \delta S=0$$

  where the end points $q_{i}\left(t_{0}\right)$ and $q_{i}\left(t_{1}\right)$ are fixed.

  Let $\phi$ and $\mathbf{A}$ be a scalar and a vector, respectively, depending on $\mathbf{r}=(x, y, z)$. Consider the Lagrangian

  $$\mathcal{L}=\frac{m \dot{\mathbf{r}}^{2}}{2}-(\phi-\dot{\mathbf{r}} \cdot \mathbf{A})$$

  and show that the resulting Euler-Lagrange equations are invariant under the transformations

  $$\phi \rightarrow \phi+\alpha \frac{\partial F}{\partial t}, \quad \mathbf{A} \rightarrow \mathbf{A}+\nabla F,$$

  where $F=F(\mathbf{r}, t)$ is an arbitrary function, and $\alpha$ is a constant which should be determined.

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• Paper 3, Section I, E

  #Classical Dynamics

  2017

  Define an integrable system with $2 n$-dimensional phase space. Define angle-action variables.

  Consider a two-dimensional phase space with the Hamiltonian

  $$H=\frac{p^{2}}{2 m}+\frac{1}{2} q^{2 k}$$

  where $k$ is a positive integer and the mass $m=m(t)$ changes slowly in time. Use the fact that the action is an adiabatic invariant to show that the energy varies in time as $m^{c}$, where $c$ is a constant which should be found.

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• Paper 4, Section I, E

  #Classical Dynamics

  2017

  Consider the Poisson bracket structure on $\mathbb{R}^{3}$ given by

  $$\{x, y\}=z, \quad\{y, z\}=x, \quad\{z, x\}=y$$

  and show that $\left\{f, \rho^{2}\right\}=0$, where $\rho^{2}=x^{2}+y^{2}+z^{2}$ and $f: \mathbb{R}^{3} \rightarrow \mathbb{R}$ is any polynomial function on $\mathbb{R}^{3}$.

  Let $H=\left(A x^{2}+B y^{2}+C z^{2}\right) / 2$, where $A, B, C$ are positive constants. Find the explicit form of Hamilton's equations

  $$\dot{\mathbf{r}}=\{\mathbf{r}, H\}, \quad \text { where } \quad \mathbf{r}=(x, y, z)$$

  Find a condition on $A, B, C$ such that the oscillation described by

  $$x=1+\alpha(t), \quad y=\beta(t), \quad z=\gamma(t)$$

  is linearly unstable, where $\alpha(t), \beta(t), \gamma(t)$ are small.

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• Paper 2, Section II, E

  #Classical Dynamics

  2017

  Show that an object's inertia tensor about a point displaced from the centre of mass by a vector $\mathbf{c}$ is given by

  $$\left(I_{\mathbf{c}}\right)_{a b}=\left(I_{0}\right)_{a b}+M\left(|\mathbf{c}|^{2} \delta_{a b}-c_{a} c_{b}\right),$$

  where $M$ is the total mass of the object, and $\left(I_{0}\right)_{a b}$ is the inertia tensor about the centre of mass.

  Find the inertia tensor of a cube of uniform density, with edge of length $L$, about one of its vertices.

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• Paper 4, Section II, $14 \mathrm{E}$

  #Classical Dynamics

  2017

  Explain how geodesics of a Riemannian metric

  $$g=g_{a b}\left(x^{c}\right) d x^{a} d x^{b}$$

  arise from the kinetic Lagrangian

  $$\mathcal{L}=\frac{1}{2} g_{a b}\left(x^{c}\right) \dot{x}^{a} \dot{x}^{b}$$

  where $a, b=1, \ldots, n$.

  Find geodesics of the metric on the upper half plane

  $$\Sigma=\left\{(x, y) \in \mathbb{R}^{2}, y>0\right\}$$

  with the metric

  $$g=\frac{d x^{2}+d y^{2}}{y^{2}}$$

  and sketch the geodesic containing the points $(2,3)$ and $(10,3)$.

  [Hint: Consider $d y / d x .]$

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• Paper 1, Section I, G

  #Coding \& Cryptography

  2017

  Let $C$ be a binary code of length $n$. Define the following decoding rules: (i) ideal observer, (ii) maximum likelihood, (iii) minimum distance.

  Let $p$ denote the probability that a digit is mistransmitted and suppose $p<1 / 2$. Prove that maximum likelihood and minimum distance decoding agree.

  Suppose codewords 000 and 111 are sent with probabilities $4 / 5$ and $1 / 5$ respectively with error probability $p=1 / 4$. If we receive 110 , how should it be decoded according to the three decoding rules above?

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• Paper 2, Section $I$, G

  #Coding \& Cryptography

  2017

  Prove that a decipherable code with prescribed word lengths exists if and only if there is a prefix-free code with the same word lengths.

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• Paper 3, Section I, G

  #Coding \& Cryptography

  2017

  Find and describe all binary cyclic codes of length 7 . Pair each code with its dual code. Justify your answer.

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• Paper 4, Section I, G

  #Coding \& Cryptography

  2017

  Describe the RSA system with public key $(N, e)$ and private key $d$.

  Give a simple example of how the system is vulnerable to a homomorphism attack.

  Describe the El-Gamal signature scheme and explain how this can defeat a homomorphism attack.

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• Paper 1, Section II, G

  #Coding \& Cryptography

  2017

  Let $C$ be a binary linear code. Explain what it means for $C$ to have length $n$ and $\operatorname{rank} k$. Explain what it means for a codeword of $C$ to have weight $j$.

  Suppose $C$ has length $n$, rank $k$, and $A_{j}$ codewords of weight $j$. The weight enumerator polynomial of $C$ is given by

  $$W_{C}(s, t)=\sum_{j=0}^{n} A_{j} s^{j} t^{n-j}$$

  What is $W_{C}(1,1) ?$ Prove that $W_{C}(s, t)=W_{C}(t, s)$ if and only if $W_{C}(1,0)=1$.

  Define the dual code $C^{\perp}$ of $C$.

  (i) Let $\mathbf{y} \in \mathbb{F}_{2}^{n}$. Show that

  $$\sum_{\mathbf{x} \in C}(-1)^{\mathbf{x} \cdot \mathbf{y}}= \begin{cases}2^{k}, & \text { if } \mathbf{y} \in C^{\perp} \\ 0, & \text { otherwise }\end{cases}$$

  (ii) Extend the definition of weight to give a weight $w(\mathbf{y})$ for $\mathbf{y} \in \mathbb{F}_{2}^{n}$. Suppose that for $t$ real and all $\mathbf{x} \in C$

  $$\sum_{\mathbf{y} \in \mathbb{F}_{2}^{n}} t^{w(\mathbf{y})}(-1)^{\mathbf{x} \cdot \mathbf{y}}=(1-t)^{w(\mathbf{x})}(1+t)^{n-w(\mathbf{x})}$$

  For $s$ real, by evaluating

  $$\sum_{\mathbf{x} \in C}\left(\sum_{\mathbf{y} \in \mathbb{F}_{2}^{n}}(-1)^{\mathbf{x} \cdot \mathbf{y}}\left(\frac{s}{t}\right)^{w(\mathbf{y})}\right)$$

  in two different ways, show that

  $$W_{C^{\perp}}(s, t)=2^{-k} W_{C}(t-s, t+s) .$$

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• Paper 2, Section II, G

  #Coding \& Cryptography

  2017

  Define the entropy, $H(X)$, of a random variable $X$. State and prove Gibbs' inequality.

  Hence, or otherwise, show that $H\left(p_{1}, p_{2}, p_{3}\right) \leqslant H\left(p_{1}, 1-p_{1}\right)+\left(1-p_{1}\right)$ and determine when equality occurs.

  Show that the Discrete Memoryless Channel with channel matrix

  $$\left(\begin{array}{ccc} 1-\alpha-\beta & \alpha & \beta \\ \alpha & 1-\alpha-\beta & \beta \end{array}\right)$$

  has capacity $C=(1-\beta)(1-\log (1-\beta))+(1-\alpha-\beta) \log (1-\alpha-\beta)+\alpha \log \alpha$.

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• Paper 1, Section I, C

  #Cosmology

  2017

  In a homogeneous and isotropic universe, describe the relative displacement $\mathbf{r}(t)$ of two galaxies in terms of a scale factor $a(t)$. Show how the relative velocity $\mathbf{v}(t)$ of these galaxies is given by the relation $\mathbf{v}(t)=H(t) \mathbf{r}(t)$, where you should specify $H(t)$ in terms of $a(t)$.

  From special relativity, the Doppler shift of light emitted by a particle moving away radially with speed $v$ can be approximated by

  $$\frac{\lambda_{0}}{\lambda_{\mathrm{e}}}=\sqrt{\frac{1+v / c}{1-v / c}}=1+\frac{v}{c}+\mathcal{O}\left(\frac{v^{2}}{c^{2}}\right)$$

  where $\lambda_{e}$ is the wavelength of emitted light and $\lambda_{0}$ is the observed wavelength. For the observed light from distant galaxies in a homogeneous and isotropic expanding universe, show that the redshift defined by $1+z \equiv \lambda_{0} / \lambda_{\mathrm{e}}$ is given by

  $$1+z=\frac{a\left(t_{0}\right)}{a\left(t_{\mathrm{e}}\right)}$$

  where $t_{\mathrm{e}}$ is the time of emission and $t_{0}$ is the observation time.

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• Paper 2, Section I, C

  #Cosmology

  2017

  In a homogeneous and isotropic universe $(\Lambda=0)$, the acceleration equation for the scale factor $a(t)$ is given by

  $$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3}\left(\rho+3 P / c^{2}\right),$$

  where $\rho(t)$ is the mass density and $P(t)$ is the pressure.

  If the matter content of the universe obeys the strong energy condition $\rho+3 P / c^{2} \geqslant 0$, show that the acceleration equation can be rewritten as $\dot{H}+H^{2} \leqslant 0$, with Hubble parameter $H(t)=\dot{a} / a$. Show that

  $$H \geqslant \frac{1}{H_{0}^{-1}+t-t_{0}}$$

  where $H_{0}=H\left(t_{0}\right)$ is the measured value today at $t=t_{0}$. Hence, or otherwise, show that

  $$a(t) \leqslant 1+H_{0}\left(t-t_{0}\right)$$

  Use this inequality to find an upper bound on the age of the universe.

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• Paper 3, Section I, C

  #Cosmology

  2017

  (a) In the early universe electrons, protons and neutral hydrogen are in thermal equilibrium and interact via,

  $$e^{-}+p^{+} \leftrightharpoons H+\gamma$$

  The non-relativistic number density of particles in thermal equlibrium is

  $$n_{i}=g_{i}\left(\frac{2 \pi m_{i} k T}{h^{2}}\right)^{\frac{3}{2}} \exp \left(\frac{\mu_{i}-m_{i} c^{2}}{k T}\right)$$

  where, for each species $i, g_{i}$ is the number of degrees of freedom, $m_{i}$ is its mass, and $\mu_{i}$ is its chemical potential. [You may assume $g_{e}=g_{p}=2$ and $g_{H}=4$.]

  Stating any assumptions required, use these expressions to derive the Saha equation which governs the relative abundances of electrons, protons and hydrogen,

  $$\frac{n_{e} n_{p}}{n_{H}}=\left(\frac{2 \pi m_{e} k T}{h^{2}}\right)^{\frac{3}{2}} \exp \left(-\frac{I}{k T}\right)$$

  where $I$ is the binding energy of hydrogen, which should be defined.

  (b) Naively, we might expect that the majority of electrons and protons combine to form neutral hydrogen once the temperature drops below the binding energy, i.e. $k T \lesssim I$. In fact recombination does not happen until a much lower temperature, when $k T \approx 0.03 I$. Briefly explain why this is.

  [Hint: It may help to consider the relative abundances of particles in the early universe.]

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• Paper 4, Section I, C

  #Cosmology

  2017

  (a) By considering a spherically symmetric star in hydrostatic equilibrium derive the pressure support equation

  $$\frac{d P}{d r}=-\frac{G M(r) \rho}{r^{2}},$$

  where $r$ is the radial distance from the centre of the star, $M(r)$ is the stellar mass contained inside that radius, and $P(r)$ and $\rho(r)$ are the pressure and density at radius $r$ respectively.

  (b) Propose, and briefly justify, boundary conditions for this differential equation, both at the centre of the star $r=0$, and at the stellar surface $r=R$.

  Suppose that $P=K \rho^{2}$ for some $K>0$. Show that the density satisfies the linear differential equation

  $$\frac{1}{x^{2}} \frac{\partial}{\partial x}\left(x^{2} \frac{\partial \rho}{\partial x}\right)=-\rho$$

  where $x=\alpha r$, for some constant $\alpha$, is a rescaled radial coordinate. Find $\alpha$.

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• Paper 3, Section II, C

  #Cosmology

  2017

  (a) The scalar moment of inertia for a system of $N$ particles is given by

  $$I=\sum_{i=1}^{N} m_{i} \mathbf{r}_{i} \cdot \mathbf{r}_{i}$$

  where $m_{i}$ is the particle's mass and $\mathbf{r}_{i}$ is a vector giving the particle's position. Show that, for non-relativistic particles,

  $$\frac{1}{2} \frac{d^{2} I}{d t^{2}}=2 K+\sum_{i=1}^{N} \mathbf{F}_{i} \cdot \mathbf{r}_{i}$$

  where $K$ is the total kinetic energy of the system and $\mathbf{F}_{i}$ is the total force on particle

  Assume that any two particles $i$ and $j$ interact gravitationally with potential energy

  $$V_{i j}=-\frac{G m_{i} m_{j}}{\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|}$$

  Show that

  $$\sum_{i=1}^{N} \mathbf{F}_{i} \cdot \mathbf{r}_{i}=V$$

  where $V$ is the total potential energy of the system. Use the above to prove the virial theorem.

  (b) Consider an approximately spherical overdensity of stationary non-interacting massive particles with initial constant density $\rho_{i}$ and initial radius $R_{i}$. Assuming the system evolves until it reaches a stable virial equilibrium, what will the final $\rho$ and $R$ be in terms of their initial values? Would this virial solution be stable if our particles were baryonic rather than non-interacting? Explain your answer.

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• Paper 1, Section II, C

  #Cosmology

  2017

  The evolution of a flat $(k=0)$ homogeneous and isotropic universe with scale factor $a(t)$, mass density $\rho(t)$ and pressure $P(t)$ obeys the Friedmann and energy conservation equations

  $$\begin{array}{r} H^{2}(t)=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G}{3} \rho+\frac{\Lambda c^{2}}{3} \\ \dot{\rho}=-3 \frac{\dot{a}}{a}\left(\rho+P / c^{2}\right) \end{array}$$

  where $H(t)$ is the Hubble parameter (observed today $t=t_{0}$ with value $H_{0}=H\left(t_{0}\right)$ ) and $\Lambda>0$ is the cosmological constant.

  Use these two equations to derive the acceleration equation

  $$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3}\left(\rho+3 P / c^{2}\right)+\frac{\Lambda c^{2}}{3}$$

  For pressure-free matter $\left(\rho=\rho_{\mathrm{M}}\right.$ and $\left.P_{\mathrm{M}}=0\right)$, solve the energy conservation equation to show that the Friedmann and acceleration equations can be re-expressed as

  $$\begin{gathered} H=H_{0} \sqrt{\frac{\Omega_{\mathrm{M}}}{a^{3}}+\Omega_{\Lambda}} \\ \frac{\ddot{a}}{a}=-\frac{H_{0}^{2}}{2}\left[\frac{\Omega_{\mathrm{M}}}{a^{3}}-2 \Omega_{\Lambda}\right] \end{gathered}$$

  where we have taken $a\left(t_{0}\right)=1$ and we have defined the relative densities today $\left(t=t_{0}\right)$ as

  $$\Omega_{\mathrm{M}}=\frac{8 \pi G}{3 H_{0}^{2}} \rho_{\mathrm{M}}\left(t_{0}\right) \quad \text { and } \quad \Omega_{\Lambda}=\frac{\Lambda c^{2}}{3 H_{0}^{2}}$$

  Solve the Friedmann equation and show that the scale factor can be expressed as

  $$a(t)=\left(\frac{\Omega_{\mathrm{M}}}{\Omega_{\Lambda}}\right)^{1 / 3} \sinh ^{2 / 3}\left(\frac{3}{2} \sqrt{\Omega_{\Lambda}} H_{0} t\right)$$

  Find an expression for the time $\bar{t}$ at which the matter density $\rho_{\mathrm{M}}$ and the effective density caused by the cosmological constant $\Lambda$ are equal. (You need not evaluate this explicitly.)

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• Paper 2, Section II, I

  #Differential Geometry

  2017

  Let $\alpha: I \rightarrow \mathbb{R}^{3}$ be a regular smooth curve. Define the curvature $k$ and torsion $\tau$ of $\alpha$ and derive the Frenet formulae. Give the assumption which must hold for torsion to be well-defined, and state the Fundamental Theorem for curves in $\mathbb{R}^{3}$.

  Let $\alpha$ be as above and $\tilde{\alpha}: I \rightarrow \mathbb{R}^{3}$ be another regular smooth curve with curvature $\tilde{k}$ and torsion $\tilde{\tau}$. Suppose $\tilde{k}(s)=k(s) \neq 0$ and $\tilde{\tau}(s)=\tau(s)$ for all $s \in I$, and that there exists a non-empty open subinterval $J \subset I$ such that $\left.\tilde{\alpha}\right|_{J}=\left.\alpha\right|_{J}$. Show that $\tilde{\alpha}=\alpha$.

  Now let $S \subset \mathbb{R}^{3}$ be an oriented surface and let $\alpha: I \rightarrow S \subset \mathbb{R}^{3}$ be a regular smooth curve contained in $S$. Define normal curvature and geodesic curvature. When is $\alpha$ a geodesic? Give an example of a surface $S$ and a geodesic $\alpha$ whose normal curvature vanishes identically. Must such a surface $S$ contain a piece of a plane? Can such a geodesic be a simple closed curve? Justify your answers.

  Show that if $\alpha$ is a geodesic and the Gaussian curvature of $S$ satisfies $K \geqslant 0$, then we have the inequality $k(s) \leqslant 2|H(\alpha(s))|$, where $H$ denotes the mean curvature of $S$ and $k$ the curvature of $\alpha$. Give an example where this inequality is sharp.

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• Paper 3, Section II, I

  #Differential Geometry

  2017

  Let $S \subset \mathbb{R}^{N}$ be a manifold and let $\alpha:[a, b] \rightarrow S \subset \mathbb{R}^{N}$ be a smooth regular curve on $S$. Define the total length $L(\alpha)$ and the arc length parameter $s$. Show that $\alpha$ can be reparametrized by arc length.

  Let $S \subset \mathbb{R}^{3}$ denote a regular surface, let $p, q \in S$ be distinct points and let $\alpha:[a, b] \rightarrow S$ be a smooth regular curve such that $\alpha(a)=p, \alpha(b)=q$. We say that $\alpha$ is length minimising if for all smooth regular curves $\tilde{\alpha}:[a, b] \rightarrow S$ with $\tilde{\alpha}(a)=p, \tilde{\alpha}(b)=q$, we have $L(\tilde{\alpha}) \geqslant L(\alpha)$. By deriving a formula for the derivative of the energy functional corresponding to a variation of $\alpha$, show that a length minimising curve is necessarily a geodesic. [You may use the following fact: given a smooth vector field $V(t)$ along $\alpha$ with $V(a)=V(b)=0$, there exists a variation $\alpha(s, t)$ of $\alpha$ such that $\left.\left.\partial_{s} \alpha(s, t)\right|_{s=0}=V(t) .\right]$

  Let $\mathbb{S}^{2} \subset \mathbb{R}^{3}$ denote the unit sphere and let $S$ denote the surface $\mathbb{S}^{2} \backslash(0,0,1)$. For which pairs of points $p, q \in S$ does there exist a length minimising smooth regular curve $\alpha:[a, b] \rightarrow S$ with $\alpha(a)=p$ and $\alpha(b)=q$ ? Justify your answer.

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• Paper 4, Section II, I

  #Differential Geometry

  2017

  Let $S \subset \mathbb{R}^{3}$ be a surface and $p \in S$. Define the exponential map exp $p$ and compute its differential $\left.d \exp _{p}\right|_{0}$. Deduce that $\exp _{p}$ is a local diffeomorphism.

  Give an example of a surface $S$ and a point $p \in S$ for which the exponential map $\exp _{p}$ fails to be defined globally on $T_{p} S$. Can this failure be remedied by extending the surface? In other words, for any such $S$, is there always a surface $S \subset \widehat{S} \subset \mathbb{R}^{3}$ such that the exponential map $\widehat{\exp }_{p}$ defined with respect to $\widehat{S}$ is globally defined on $T_{p} S=T_{p} \widehat{S}$ ?

  State the version of the Gauss-Bonnet theorem with boundary term for a surface $S \subset \mathbb{R}^{3}$ and a closed disc $D \subset S$ whose boundary $\partial D$ can be parametrized as a smooth closed curve in $S$.

  Let $S \subset \mathbb{R}^{3}$ be a flat surface, i.e. $K=0$. Can there exist a closed disc $D \subset S$, whose boundary $\partial D$ can be parametrized as a smooth closed curve, and a surface $\tilde{S} \subset \mathbb{R}^{3}$ such that all of the following hold:

  (i) $(S \backslash D) \cup \partial D \subset \tilde{S}$;

  (ii) letting $\tilde{D}$ be $(\tilde{S} \backslash(S \backslash D)) \cup \partial D$, we have that $\tilde{D}$ is a closed disc in $\tilde{S}$ with boundary $\partial \tilde{D}=\partial D$

  (iii) the Gaussian curvature $\tilde{K}$ of $\tilde{S}$ satisfies $\tilde{K} \geqslant 0$, and there exists a $p \in \tilde{S}$ such that $\tilde{K}(p)>0$ ?

  Justify your answer.

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• Paper 1, Section II, I

  #Differential Geometry

  2017

  Define what it means for a subset $X \subset \mathbb{R}^{N}$ to be a manifold.

  For manifolds $X$ and $Y$, state what it means for a map $f: X \rightarrow Y$ to be smooth. For such a smooth map, and $x \in X$, define the differential map $d f_{x}$.

  What does it mean for $y \in Y$ to be a regular value of $f$ ? Give an example of a map $f: X \rightarrow Y$ and a $y \in Y$ which is not a regular value of $f$.

  Show that the set $S L_{n}(\mathbb{R})$ of $n \times n$ real-valued matrices with determinant 1 can naturally be viewed as a manifold $S L_{n}(\mathbb{R}) \subset \mathbb{R}^{n^{2}}$. What is its dimension? Show that matrix multiplication $f: S L_{n}(\mathbb{R}) \times S L_{n}(\mathbb{R}) \rightarrow S L_{n}(\mathbb{R})$, defined by $f(A, B)=A B$, is smooth. [Standard theorems may be used without proof if carefully stated.] Describe the tangent space of $S L_{n}(\mathbb{R})$ at the identity $I \in S L_{n}(\mathbb{R})$ as a subspace of $\mathbb{R}^{n^{2}}$.

  Show that if $n \geqslant 2$ then the set of real-valued matrices with determinant 0 , viewed as a subset of $\mathbb{R}^{n^{2}}$, is not a manifold.

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• Paper 1, Section II, A

  #Dynamical Systems

  2017

  Consider the dynamical system

  $$\begin{aligned} &\dot{x}=-x+x^{3}+\beta x y^{2} \\ &\dot{y}=-y+\beta x^{2} y+y^{3} \end{aligned}$$

  where $\beta>-1$ is a constant.

  (a) Find the fixed points of the system, and classify them for $\beta \neq 1$.

  Sketch the phase plane for each of the cases (i) $\beta=\frac{1}{2}$ (ii) $\beta=2$ and (iii) $\beta=1$.

  (b) Given $\beta>2$, show that the domain of stability of the origin includes the union over $k \in \mathbb{R}$ of the regions

  $$x^{2}+k^{2} y^{2}<\frac{4 k^{2}\left(1+k^{2}\right)(\beta-1)}{\beta^{2}\left(1+k^{2}\right)^{2}-4 k^{2}} .$$

  By considering $k \gg 1$, or otherwise, show that more information is obtained from the union over $k$ than considering only the case $k=1$.

  $\left[\right.$ Hint: If $B>A, C$ then $\left.\max _{u \in[0,1]}\left\{A u^{2}+2 B u(1-u)+C(1-u)^{2}\right\}=\frac{B^{2}-A C}{2 B-A-C} \cdot\right]$

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• Paper 2, Section II, A

  #Dynamical Systems

  2017

  (a) State Liapunov's first theorem and La Salle's invariance principle. Use these results to show that the fixed point at the origin of the system

  $$\ddot{x}+k \dot{x}+\sin ^{3} x=0, \quad k>0,$$

  is asymptotically stable.

  (b) State the Poincaré-Bendixson theorem. Show that the forced damped pendulum

  $$\dot{\theta}=p, \quad \dot{p}=-k p-\sin \theta+F, \quad k>0,$$

  with $F>1$, has a periodic orbit that encircles the cylindrical phase space $(\theta, p) \in \mathbb{R}[\bmod 2 \pi] \times \mathbb{R}$, and that it is unique.

  [You may assume that the Poincaré-Bendixson theorem also holds on a cylinder, and comment, without proof, on the use of any other standard results.]

  (c) Now consider $(*)$ for $F, k=O(\epsilon)$, where $\epsilon \ll 1$. Use the energy-balance method to show that there is a homoclinic orbit in $p \geqslant 0$ if $F=F_{h}(k)$, where $F_{h} \approx 4 k / \pi>0$.

  Explain briefly why there is no homoclinic orbit in $p \leqslant 0$ for $F>0$.

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• Paper 3, Section II, A

  #Dynamical Systems

  2017

  State, without proof, the centre manifold theorem. Show that the fixed point at the origin of the system

  $$\begin{aligned} &\dot{x}=y-x+a x^{3} \\ &\dot{y}=r x-y-y z \\ &\dot{z}=x y-z \end{aligned}$$

  where $a \neq 1$ is a constant, is nonhyperbolic at $r=1$. What are the dimensions of the linear stable and (non-extended) centre subspaces at this point?

  Make the substitutions $2 u=x+y, 2 v=x-y$ and $\mu=r-1$ and derive the resultant equations for $\dot{u}, \dot{v}$ and $\dot{z}$.

  The extended centre manifold is given by

  $$v=V(u, \mu), \quad z=Z(u, \mu),$$

  where $V$ and $Z$ can be expanded as power series about $u=\mu=0$. What is known about $V$ and $Z$ from the centre manifold theorem? Assuming that $\mu=O\left(u^{2}\right)$, determine $Z$ to $O\left(u^{2}\right)$ and $V$ to $O\left(u^{3}\right)$. Hence obtain the evolution equation on the centre manifold correct to $O\left(u^{3}\right)$, and identify the type of bifurcation distinguishing between the cases $a>1$ and $a<1$.

  If now $a=1$, assume that $\mu=O\left(u^{4}\right)$ and extend your calculations of $Z$ to $O\left(u^{4}\right)$ and of the dynamics on the centre manifold to $O\left(u^{5}\right)$. Hence sketch the bifurcation diagram in the neighbourhood of $u=\mu=0$.

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• Paper 4, Section II, A

  #Dynamical Systems

  2017

  Consider the one-dimensional map $F: \mathbb{R} \rightarrow \mathbb{R}$ defined by

  $$x_{i+1}=F\left(x_{i} ; \mu\right)=x_{i}\left(a x_{i}^{2}+b x_{i}+\mu\right),$$

  where $a$ and $b$ are constants, $\mu$ is a parameter and $a \neq 0$.

  (a) Find the fixed points of $F$ and determine the linear stability of $x=0$. Hence show that there are bifurcations at $\mu=1$, at $\mu=-1$ and, if $b \neq 0$, at $\mu=1+b^{2} /(4 a)$.

  Sketch the bifurcation diagram for each of the cases:

  $$\text { (i) } a>b=0, \quad \text { (ii) } a, b>0 \text { and (iii) } a, b<0 \text {. }$$

  In each case show the locus and stability of the fixed points in the $(\mu, x)$-plane, and state the type of each bifurcation. [Assume that there are no further bifurcations in the region sketched.]

  (b) For the case $F(x)=x\left(\mu-x^{2}\right)$ (i.e. $\left.a=-1, b=0\right)$, you may assume that

  $$F^{2}(x)=x+x\left(\mu-1-x^{2}\right)\left(\mu+1-x^{2}\right)\left(1-\mu x^{2}+x^{4}\right) .$$

  Show that there are at most three 2-cycles and determine when they exist. By considering $F^{\prime}\left(x_{i}\right) F^{\prime}\left(x_{i+1}\right)$, or otherwise, show further that one 2-cycle is always unstable when it exists and that the others are unstable when $\mu>\sqrt{5}$. Sketch the bifurcation diagram showing the locus and stability of the fixed points and 2-cycles. State briefly what you would expect to occur for $\mu>\sqrt{5}$.

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• Paper 1, Section II, 35D

  #Electrodynamics

  2017

  In some inertial reference frame $S$, there is a uniform electric field $\mathbf{E}$ directed along the positive $y$-direction and a uniform magnetic field $\mathbf{B}$ directed along the positive $z$ direction. The magnitudes of the fields are $E$ and $B$, respectively, with $E<c B$. Show that it is possible to find a reference frame in which the electric field vanishes, and determine the relative speed $\beta c$ of the two frames and the magnitude of the magnetic field in the new frame.

  [Hint: You may assume that under a standard Lorentz boost with speed $v=\beta$ c along the $x$-direction, the electric and magnetic field components transform as

  $$\left(\begin{array}{c} E_{x}^{\prime} \\ E_{y}^{\prime} \\ E_{z}^{\prime} \end{array}\right)=\left(\begin{array}{c} E_{x} \\ \gamma(\beta)\left(E_{y}-v B_{z}\right) \\ \gamma(\beta)\left(E_{z}+v B_{y}\right) \end{array}\right) \quad a n d \quad\left(\begin{array}{c} B_{x}^{\prime} \\ B_{y}^{\prime} \\ B_{z}^{\prime} \end{array}\right)=\left(\begin{array}{c} B_{x} \\ \gamma(\beta)\left(B_{y}+v E_{z} / c^{2}\right) \\ \gamma(\beta)\left(B_{z}-v E_{y} / c^{2}\right) \end{array}\right),$$

  where the Lorentz factor $\gamma(\beta)=\left(1-\beta^{2}\right)^{-1 / 2}$.]

  A point particle of mass $m$ and charge $q$ moves relativistically under the influence of the fields $\mathbf{E}$ and $\mathbf{B}$. The motion is in the plane $z=0$. By considering the motion in the reference frame in which the electric field vanishes, or otherwise, show that, with a suitable choice of origin, the worldline of the particle has components in the frame $S$ of the form

  $$\begin{aligned} &c t(\tau)=\gamma(u / c) \gamma(\beta)\left[c \tau+\frac{\beta u}{\omega} \sin \omega \tau\right] \\ &x(\tau)=\gamma(u / c) \gamma(\beta)\left[\beta c \tau+\frac{u}{\omega} \sin \omega \tau\right] \\ &y(\tau)=\frac{u \gamma(u / c)}{\omega} \cos \omega \tau \end{aligned}$$

  Here, $u$ is a constant speed with Lorentz factor $\gamma(u / c), \tau$ is the particle's proper time, and $\omega$ is a frequency that you should determine.

  Using dimensionless coordinates,

  $$\tilde{x}=\frac{\omega}{u \gamma(u / c)} x \quad \text { and } \quad \tilde{y}=\frac{\omega}{u \gamma(u / c)} y$$

  sketch the trajectory of the particle in the $(\tilde{x}, \tilde{y})$-plane in the limiting cases $2 \pi \beta \ll u / c$ and $2 \pi \beta \gg u / c$.

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• Paper 3, Section II, D

  #Electrodynamics

  2017

  By considering the force per unit volume $\mathbf{f}=\rho \mathbf{E}+\mathbf{J} \times \mathbf{B}$ on a charge density $\rho$ and current density $\mathbf{J}$ due to an electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$, show that

  $$\frac{\partial g_{i}}{\partial t}+\frac{\partial \sigma_{i j}}{\partial x_{j}}=-f_{i}$$

  where $\mathbf{g}=\epsilon_{0} \mathbf{E} \times \mathbf{B}$ and the symmetric tensor $\sigma_{i j}$ should be specified.

  Give the physical interpretation of $\mathbf{g}$ and $\sigma_{i j}$ and explain how $\sigma_{i j}$ can be used to calculate the net electromagnetic force exerted on the charges and currents within some region of space in static situations.

  The plane $x=0$ carries a uniform charge $\sigma$ per unit area and a current $K$ per unit length along the $z$-direction. The plane $x=d$ carries the opposite charge and current. Show that between these planes

  $$\sigma_{i j}=\frac{\sigma^{2}}{2 \epsilon_{0}}\left(\begin{array}{ccc} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)+\frac{\mu_{0} K^{2}}{2}\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{array}\right)$$

  and $\sigma_{i j}=0$ for $x<0$ and $x>d$.

  Use $(*)$ to find the electromagnetic force per unit area exerted on the charges and currents in the $x=0$ plane. Show that your result agrees with direct calculation of the force per unit area based on the Lorentz force law.

  If the current $K$ is due to the motion of the charge $\sigma$ with speed $v$, is it possible for the force between the planes to be repulsive?

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• Paper 4, Section II, D

  #Electrodynamics

  2017

  A dielectric material has a real, frequency-independent relative permittivity $\epsilon_{r}$ with $\left|\epsilon_{r}-1\right| \ll 1$. In this case, the macroscopic polarization that develops when the dielectric is placed in an external electric field $\mathbf{E}_{\text {ext }}(t, \mathbf{x})$ is $\mathbf{P}(t, \mathbf{x}) \approx \epsilon_{0}\left(\epsilon_{r}-1\right) \mathbf{E}_{\text {ext }}(t, \mathbf{x})$. Explain briefly why the associated bound current density is

  $$\mathbf{J}_{\text {bound }}(t, \mathbf{x}) \approx \epsilon_{0}\left(\epsilon_{r}-1\right) \frac{\partial \mathbf{E}_{\text {ext }}(t, \mathbf{x})}{\partial t}$$

  [You should ignore any magnetic response of the dielectric.]

  A sphere of such a dielectric, with radius $a$, is centred on $\mathbf{x}=0$. The sphere scatters an incident plane electromagnetic wave with electric field

  $$\mathbf{E}(t, \mathbf{x})=\mathbf{E}_{0} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}$$

  where $\omega=c|\mathbf{k}|$ and $\mathbf{E}_{0}$ is a constant vector. Working in the Lorenz gauge, show that at large distances $r=|\mathbf{x}|$, for which both $r \gg a$ and $k a^{2} / r \ll 2 \pi$, the magnetic vector potential $\mathbf{A}_{\text {scatt }}(t, \mathbf{x})$ of the scattered radiation is

  $$\mathbf{A}_{\mathrm{scatt}}(t, \mathbf{x}) \approx-i \omega \mathbf{E}_{0} \frac{e^{i(k r-\omega t)}}{r} \frac{\left(\epsilon_{r}-1\right)}{4 \pi c^{2}} \int_{\left|\mathbf{x}^{\prime}\right| \leqslant a} e^{i \mathbf{q} \cdot \mathbf{x}^{\prime}} d^{3} \mathbf{x}^{\prime}$$

  where $\mathbf{q}=\mathbf{k}-k \hat{\mathbf{x}}$ with $\hat{\mathbf{x}}=\mathbf{x} / r$.

  In the far-field, where $k r \gg 1$, the electric and magnetic fields of the scattered radiation are given by

  $$\begin{aligned} &\mathbf{E}_{\text {scatt }}(t, \mathbf{x}) \approx-i \omega \hat{\mathbf{x}} \times\left[\hat{\mathbf{x}} \times \mathbf{A}_{\text {scatt }}(t, \mathbf{x})\right] \\ &\mathbf{B}_{\text {scatt }}(t, \mathbf{x}) \approx i k \hat{\mathbf{x}} \times \mathbf{A}_{\text {scatt }}(t, \mathbf{x}) \end{aligned}$$

  By calculating the Poynting vector of the scattered and incident radiation, show that the ratio of the time-averaged power scattered per unit solid angle to the time-averaged incident power per unit area (i.e. the differential cross-section) is

  $$\frac{d \sigma}{d \Omega}=\left(\epsilon_{r}-1\right)^{2} k^{4}\left(\frac{\sin (q a)-q a \cos (q a)}{q^{3}}\right)^{2}\left|\hat{\mathbf{x}} \times \hat{\mathbf{E}}_{0}\right|^{2}$$

  where $\hat{\mathbf{E}}_{0}=\mathbf{E}_{0} /\left|\mathbf{E}_{0}\right|$ and $q=|\mathbf{q}|$.

  [You may assume that, in the Lorenz gauge, the retarded potential due to a localised current distribution is

  $$\mathbf{A}(t, \mathbf{x})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(t_{\mathrm{ret}}, \mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} d^{3} \mathbf{x}^{\prime},$$

  where the retarded time $\left.t_{\text {ret }}=t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right| / c .\right]$

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• Paper 2, Section II, B

  #Fluid Dynamics II

  2017

  A cylinder of radius $a$ falls at speed $U$ without rotating through viscous fluid adjacent to a vertical plane wall, with its axis horizontal and parallel to the wall. The distance between the cylinder and the wall is $h_{0} \ll a$. Use lubrication theory in a frame of reference moving with the cylinder to determine that the two-dimensional volume flux between the cylinder and the wall is

  $$q=\frac{2 h_{0} U}{3}$$

  upwards, relative to the cylinder.

  Determine an expression for the viscous shear stress on the cylinder. Use this to calculate the viscous force and hence the torque on the cylinder. If the cylinder is free to rotate, what does your result say about the sense of rotation of the cylinder?

  [Hint: You may quote the following integrals:

  $$\left.\int_{-\infty}^{\infty} \frac{d t}{1+t^{2}}=\pi, \quad \int_{-\infty}^{\infty} \frac{d t}{\left(1+t^{2}\right)^{2}}=\frac{\pi}{2}, \quad \int_{-\infty}^{\infty} \frac{d t}{\left(1+t^{2}\right)^{3}}=\frac{3 \pi}{8}\right]$$

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• Paper 1, Section II, B

  #Fluid Dynamics II

  2017

  Fluid of density $\rho$ and dynamic viscosity $\mu$ occupies the region $y>0$ in Cartesian coordinates $(x, y, z)$. A semi-infinite, dense array of cilia occupy the half plane $y=0$, $x>0$ and apply a stress in the $x$-direction on the adjacent fluid, working at a constant and uniform rate $\rho P$ per unit area, which causes the fluid to move with steady velocity $\mathbf{u}=(u(x, y), v(x, y), 0)$. Give a careful physical explanation of the boundary condition

  $$\left.u \frac{\partial u}{\partial y}\right|_{y=0}=-\frac{P}{\nu} \quad \text { for } \quad x>0$$

  paying particular attention to signs, where $\nu$ is the kinematic viscosity of the fluid. Why would you expect the fluid motion to be confined to a thin region near $y=0$ for sufficiently large values of $x$ ?

  Write down the viscous-boundary-layer equations governing the thin region of fluid motion. Show that the flow can be approximated by a stream function

  $$\psi(x, y)=U(x) \delta(x) f(\eta), \quad \text { where } \quad \eta=\frac{y}{\delta(x)}$$

  Determine the functions $U(x)$ and $\delta(x)$. Show that the dimensionless function $f(\eta)$ satisfies

  $$f^{\prime \prime \prime}=\frac{1}{5} f^{\prime 2}-\frac{3}{5} f f^{\prime \prime}$$

  What boundary conditions must be satisfied by $f(\eta)$ ? By considering how the volume flux varies with downstream location $x$, or otherwise, determine (with justification) the sign of the transverse flow $v$.

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• Paper 3, Section II, B

  #Fluid Dynamics II

  2017

  A spherical bubble of radius a moves with velocity $\mathbf{U}$ through a viscous fluid that is at rest far from the bubble. The pressure and velocity fields outside the bubble are given by

  $$p=\mu \frac{a}{r^{3}} \mathbf{U} \cdot \mathbf{x} \quad \text { and } \quad \mathbf{u}=\frac{a}{2 r} \mathbf{U}+\frac{a}{2 r^{3}}(\mathbf{U} \cdot \mathbf{x}) \mathbf{x}$$

  respectively, where $\mu$ is the dynamic viscosity of the fluid, $\mathbf{x}$ is the position vector from the centre of the bubble and $r=|\mathbf{x}|$. Using suffix notation, or otherwise, show that these fields satisfy the Stokes equations.

  Obtain an expression for the stress tensor for the fluid outside the bubble and show that the velocity field above also satisfies all the appropriate boundary conditions.

  Compute the drag force on the bubble.

  [Hint: You may use

  $$\int_{S} n_{i} n_{j} d S=\frac{4}{3} \pi a^{2} \delta_{i j}$$

  where the integral is taken over the surface of a sphere of radius a and $\mathbf{n}$ is the outward unit normal to the surface.]

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• Paper 4, Section II, B

  #Fluid Dynamics II

  2017

  A horizontal layer of inviscid fluid of density $\rho_{1}$ occupying $0<y<h$ flows with velocity $(U, 0)$ above a horizontal layer of inviscid fluid of density $\rho_{2}>\rho_{1}$ occupying $-h<y<0$ and flowing with velocity $(-U, 0)$, in Cartesian coordinates $(x, y)$. There are rigid boundaries at $y=\pm h$. The interface between the two layers is perturbed to position $y=\operatorname{Re}\left(A e^{i k x+\sigma t}\right)$.

  Write down the full set of equations and boundary conditions governing this flow. Derive the linearised boundary conditions appropriate in the limit $A \rightarrow 0$. Solve the linearised equations to show that the perturbation to the interface grows exponentially in time if

  $$U^{2}>\frac{\rho_{2}^{2}-\rho_{1}^{2}}{\rho_{1} \rho_{2}} \frac{g}{4 k} \tanh k h .$$

  Sketch the right-hand side of this inequality as a function of $k$. Thereby deduce the minimum value of $U$ that makes the system unstable for all wavelengths.

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• Paper 1, Section I, E

  #Further Complex Methods

  2017

  Calculate the value of the integral

  $$P \int_{-\infty}^{\infty} \frac{e^{-i x}}{x^{n}} d x$$

  where $P$ stands for Principal Value and $n$ is a positive integer.

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• Paper 2, Section I, E

  #Further Complex Methods

  2017

  Euler's formula for the Gamma function is

  $$\Gamma(z)=\frac{1}{z} \prod_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{z}\left(1+\frac{z}{n}\right)^{-1}$$

  Use Euler's formula to show

  $$\frac{\Gamma(2 z)}{2^{2 z} \Gamma(z) \Gamma\left(z+\frac{1}{2}\right)}=C$$

  where $C$ is a constant.

  Evaluate $C$.

  [Hint: You may use $\Gamma(z) \Gamma(1-z)=\pi / \sin (\pi z) .]$

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• Paper 3, Section I, E

  #Further Complex Methods

  2017

  Find all the singular points of the differential equation

  $$z \frac{d^{2} y}{d z^{2}}+(2-z) \frac{d y}{d z}-y=0$$

  and determine whether they are regular or irregular singular points.

  By writing $y(z)=f(z) / z$, find two linearly independent solutions to this equation.

  Comment on the relationship of your solutions to the nature of the singular points of the original differential equation.

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• Paper 4, Section I, $7 \mathbf{E} \quad$

  #Further Complex Methods

  2017

  Consider the differential equation

  $$z \frac{d^{2} y}{d z^{2}}-2 \frac{d y}{d z}+z y=0$$

  Laplace's method finds a solution of this differential equation by writing $y(z)$ in the form

  $$y(z)=\int_{C} e^{z t} f(t) d t$$

  where $C$ is a closed contour.

  Determine $f(t)$. Hence find two linearly independent real solutions of $(\star)$ for $z$ real.

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• Paper 2, Section II, E

  #Further Complex Methods

  2017

  The hypergeometric equation is represented by the Papperitz symbol

  $$P\left\{\begin{array}{ccc} 0 & 1 & \infty \\ 0 & 0 & a \\ 1-c & c-a-b & b \end{array}\right\}$$

  and has solution $y_{0}(z)=F(a, b, c ; z)$.

  Functions $y_{1}(z)$ and $y_{2}(z)$ are defined by

  $$y_{1}(z)=F(a, b, a+b+1-c ; 1-z)$$

  and

  $$y_{2}(z)=(1-z)^{c-a-b} F(c-a, c-b, c-a-b+1 ; 1-z),$$

  where $c-a-b$ is not an integer.

  Show that $y_{1}(z)$ and $y_{2}(z)$ obey the hypergeometric equation $(*)$.

  Explain why $y_{0}(z)$ can be written in the form

  $$y_{0}(z)=A y_{1}(z)+B y_{2}(z)$$

  where $A$ and $B$ are independent of $z$ but depend on $a, b$ and $c$.

  Suppose that

  $$F(a, b, c ; z)=\frac{\Gamma(c)}{\Gamma(b) \Gamma(c-b)} \int_{0}^{1} t^{b-1}(1-t)^{c-b-1}(1-t z)^{-a} d t$$

  with $\operatorname{Re}(c)>\operatorname{Re}(b)>0$ and $|\arg (1-z)|<\pi$. Find expressions for $A$ and $B$.

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• Paper 1, Section II, E

  #Further Complex Methods

  2017

  The Riemann zeta function is defined by

  $$\zeta_{R}(s)=\sum_{n=1}^{\infty} n^{-s}$$

  for $\operatorname{Re}(s)>1$.

  Show that

  $$\zeta_{R}(s)=\frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{t^{s-1}}{e^{t}-1} d t$$

  Let $I(s)$ be defined by

  $$I(s)=\frac{\Gamma(1-s)}{2 \pi i} \int_{C} \frac{t^{s-1}}{e^{-t}-1} d t$$

  where $C$ is the Hankel contour.

  Show that $I(s)$ provides an analytic continuation of $\zeta_{R}(s)$ for a range of $s$ which should be determined.

  Hence evaluate $\zeta_{R}(-1)$.

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• Paper 2, Section II, I

  #Galois Theory

  2017

  (a) Define what it means for a finite field extension $L$ of a field $K$ to be separable. Show that $L$ is of the form $K(\alpha)$ for some $\alpha \in L$.

  (b) Let $p$ and $q$ be distinct prime numbers. Let $L=\mathbb{Q}(\sqrt{p}, \sqrt{-q})$. Express $L$ in the form $\mathbb{Q}(\alpha)$ and find the minimal polynomial of $\alpha$ over $\mathbb{Q}$.

  (c) Give an example of a field extension $K \leqslant L$ of finite degree, where $L$ is not of the form $K(\alpha)$. Justify your answer.

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• Paper 3, Section II, I

  #Galois Theory

  2017

  (a) Let $F$ be a finite field of characteristic $p$. Show that $F$ is a finite Galois extension of the field $F_{p}$ of $p$ elements, and that the Galois group of $F$ over $F_{p}$ is cyclic.

  (b) Find the Galois groups of the following polynomials:

  (i) $t^{4}+1$ over $F_{3}$.

  (ii) $t^{3}-t-2$ over $F_{5}$.

  (iii) $t^{4}-1$ over $F_{7}$.

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• Paper 1, Section II, I

  #Galois Theory

  2017

  (a) Let $K$ be a field and let $f(t) \in K[t]$. What does it mean for a field extension $L$ of $K$ to be a splitting field for $f(t)$ over $K$ ?

  Show that the splitting field for $f(t)$ over $K$ is unique up to isomorphism.

  (b) Find the Galois groups over the rationals $\mathbb{Q}$ for the following polynomials: (i) $t^{4}+2 t+2$. (ii) $t^{5}-t-1$.

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• Paper 4, Section II, I

  #Galois Theory

  2017

  (a) State the Fundamental Theorem of Galois Theory.

  (b) What does it mean for an extension $L$ of $\mathbb{Q}$ to be cyclotomic? Show that a cyclotomic extension $L$ of $\mathbb{Q}$ is a Galois extension and prove that its Galois group is Abelian.

  (c) What is the Galois group $G$ of $\mathbb{Q}(\eta)$ over $\mathbb{Q}$, where $\eta$ is a primitive 7 th root of unity? Identify the intermediate subfields $M$, with $\mathbb{Q} \leqslant M \leqslant \mathbb{Q}(\eta)$, in terms of $\eta$, and identify subgroups of $G$ to which they correspond. Justify your answers.

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• Paper 2, Section II, D

  #General Relativity

  2017

  (a) The Friedmann-Robertson-Walker metric is given by

  $$d s^{2}=-d t^{2}+a^{2}(t)\left[\frac{d r^{2}}{1-k r^{2}}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right]$$

  where $k=-1,0,+1$ and $a(t)$ is the scale factor.

  For $k=+1$, show that this metric can be written in the form

  $$d s^{2}=-d t^{2}+\gamma_{i j} d x^{i} d x^{j}=-d t^{2}+a^{2}(t)\left[d \chi^{2}+\sin ^{2} \chi\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right]$$

  Calculate the equatorial circumference $(\theta=\pi / 2)$ of the submanifold defined by constant $t$ and $\chi$.

  Calculate the proper volume, defined by $\int \sqrt{\operatorname{det} \gamma} d^{3} x$, of the hypersurface defined by constant $t$.

  (b) The Friedmann equations are

  $$\begin{aligned} &3\left(\frac{\dot{a}^{2}+k}{a^{2}}\right)-\Lambda=8 \pi \rho, \\ &\frac{2 a \ddot{a}+\dot{a}^{2}+k}{a^{2}}-\Lambda=-8 \pi P, \end{aligned}$$

  where $\rho(t)$ is the energy density, $P(t)$ is the pressure, $\Lambda$ is the cosmological constant and dot denotes $d / d t$.

  The Einstein static universe has vanishing pressure, $P(t)=0$. Determine $a, k$ and $\Lambda$ as a function of the density $\rho$.

  The Einstein static universe with $a=a_{0}$ and $\rho=\rho_{0}$ is perturbed by radiation such that

  $$a=a_{0}+\delta a(t), \quad \rho=\rho_{0}+\delta \rho(t), \quad P=\frac{1}{3} \delta \rho(t)$$

  where $\delta a \ll a_{0}$ and $\delta \rho \ll \rho_{0}$. Show that the Einstein static universe is unstable to this perturbation.

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• Paper 1, Section II, D

  #General Relativity

  2017

  A static black hole in a five-dimensional spacetime is described by the metric

  $$d s^{2}=-\left(1-\frac{\mu}{r^{2}}\right) d t^{2}+\left(1-\frac{\mu}{r^{2}}\right)^{-1} d r^{2}+r^{2}\left[d \psi^{2}+\sin ^{2} \psi\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right]$$

  where $\mu>0$ is a constant.

  A geodesic lies in the plane $\theta=\psi=\pi / 2$ and has affine parameter $\lambda$. Show that

  $$E=\left(1-\frac{\mu}{r^{2}}\right) \frac{d t}{d \lambda} \quad \text { and } \quad L=r^{2} \frac{d \phi}{d \lambda}$$

  are both constants of motion. Write down a third constant of motion.

  Show that timelike and null geodesics satisfy the equation

  $$\frac{1}{2}\left(\frac{d r}{d \lambda}\right)^{2}+V(r)=\frac{1}{2} E^{2}$$

  for some potential $V(r)$ which you should determine.

  Circular geodesics satisfy the equation $V^{\prime}(r)=0$. Calculate the values of $r$ for which circular null geodesics exist and for which circular timelike geodesics exist. Which are stable and which are unstable? Briefly describe how this compares to circular geodesics in the four-dimensional Schwarzschild geometry.

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• Paper 3, Section II, D

  #General Relativity

  2017

  Let $\mathcal{M}$ be a two-dimensional manifold with metric $\boldsymbol{g}$ of signature $-+$.

  (i) Let $p \in \mathcal{M}$. Use normal coordinates at the point $p$ to show that one can choose two null vectors $\mathbf{V}, \mathbf{W}$ that form a basis of the vector space $\mathcal{T}_{p}(\mathcal{M})$.

  (ii) Consider the interval $I \subset \mathbb{R}$. Let $\gamma: I \rightarrow \mathcal{M}$ be a null curve through $p$ and $\mathbf{U} \neq 0$ be the tangent vector to $\gamma$ at $p$. Show that the vector $\mathbf{U}$ is either parallel to $\mathbf{V}$ or parallel to $\mathbf{W}$.

  (iii) Show that every null curve in $\mathcal{M}$ is a null geodesic.

  [Hint: You may wish to consider the acceleration $a^{\alpha}=U^{\beta} \nabla_{\beta} U^{\alpha}$.]

  (iv) By providing an example, show that not every null curve in four-dimensional Minkowski spacetime is a null geodesic.

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• Paper 4, Section II, D

  #General Relativity

  2017

  (a) In the transverse traceless gauge, a plane gravitational wave propagating in the $z$ direction is described by a perturbation $h_{\alpha \beta}$ of the Minkowski metric $\eta_{\alpha \beta}=$ $\operatorname{diag}(-1,1,1,1)$ in Cartesian coordinates $x^{\alpha}=(t, x, y, z)$, where

  $$h_{\alpha \beta}=H_{\alpha \beta} e^{i k_{\mu} x^{\mu}}, \quad \text { where } \quad k^{\mu}=\omega(1,0,0,1)$$

  and $H_{\alpha \beta}$ is a constant matrix. Spacetime indices in this question are raised or lowered with the Minkowski metric.

  The energy-momentum tensor of a gravitational wave is defined to be

  $$\tau_{\mu \nu}=\frac{1}{32 \pi}\left(\partial_{\mu} h^{\alpha \beta}\right)\left(\partial_{\nu} h_{\alpha \beta}\right)$$

  Show that $\partial^{\nu} \tau_{\mu \nu}=\frac{1}{2} \partial_{\mu} \tau_{\nu}^{\nu}$ and hence, or otherwise, show that energy and momentum are conserved.

  (b) A point mass $m$ undergoes harmonic motion along the $z$-axis with frequency $\omega$ and amplitude $L$. Compute the energy flux emitted in gravitational radiation.

  [Hint: The quadrupole formula for time-averaged energy flux radiated in gravitational waves is

  \left\langle\frac{d E}{d t}\right\rangle=\frac{1}{5}\left\langle\dddot{Q}_{i j} \dddot{Q}_{i j}\right\rangle

  where $Q_{i j}$ is the reduced quadrupole tensor.]

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• Paper 3, Section II, H

  #Graph Theory

  2017

  Define the Ramsey numbers $R(s, t)$ for integers $s, t \geqslant 2$. Show that $R(s, t)$ exists for all $s, t \geqslant 2$. Show also that $R(s, s) \leqslant 4^{s}$ for all $s \geqslant 2$.

  Let $t \geqslant 2$ be fixed. Give a red-blue colouring of the edges of $K_{2 t-2}$ for which there is no red $K_{t}$ and no blue odd cycle. Show, however, that for any red-blue colouring of the edges of $K_{2 t-1}$ there must exist either a red $K_{t}$ or a blue odd cycle.

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• Paper 2, Section II, H

  #Graph Theory

  2017

  State and prove Hall's theorem about matchings in bipartite graphs.

  Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix, with all entries non-negative reals, such that every row sum and every column sum is 1. By applying Hall's theorem, show that there is a permutation $\sigma$ of $\{1, \ldots, n\}$ such that $a_{i \sigma(i)}>0$ for all $i$.

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• Paper 1, Section II, H

  #Graph Theory

  2017

  Let $G$ be a graph of order $n \geqslant 3$ satisfying $\delta(G) \geqslant \frac{n}{2}$. Show that $G$ is Hamiltonian.

  Give an example of a planar graph $G$, with $\chi(G)=4$, that is Hamiltonian, and also an example of a planar graph $G$, with $\chi(G)=4$, that is not Hamiltonian.

  Let $G$ be a planar graph with the property that the boundary of the unbounded face is a Hamilton cycle of $G$. Prove that $\chi(G) \leqslant 3$.

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• Paper 4, Section II, H

  #Graph Theory

  2017

  Let $G$ be a graph of maximum degree $\Delta$. Show the following:

  (i) Every eigenvalue $\lambda$ of $G$ satisfies $|\lambda| \leqslant \Delta$.

  (ii) If $G$ is regular then $\Delta$ is an eigenvalue.

  (iii) If $G$ is regular and connected then the multiplicity of $\Delta$ as an eigenvalue is 1 .

  (iv) If $G$ is regular and not connected then the multiplicity of $\Delta$ as an eigenvalue is greater than 1 .

  Let $A$ be the adjacency matrix of the Petersen graph. Explain why $A^{2}+A-2 I=J$, where $I$ is the identity matrix and $J$ is the all-1 matrix. Find, with multiplicities, the eigenvalues of the Petersen graph.

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• Paper 1, Section II, A

  #Integrable Systems

  2017

  Define a Lie point symmetry of the first order ordinary differential equation $\Delta[t, \mathbf{x}, \dot{\mathbf{x}}]=$ 0. Describe such a Lie point symmetry in terms of the vector field that generates it.

  Consider the $2 n$-dimensional Hamiltonian system $(M, H)$ governed by the differential equation

  $$\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t}=J \frac{\partial H}{\partial \mathbf{x}}$$

  Define the Poisson bracket $\{\cdot, \cdot\}$. For smooth functions $f, g: M \rightarrow \mathbf{R}$ show that the associated Hamiltonian vector fields $V_{f}, V_{g}$ satisfy

  $$\left[V_{f}, V_{g}\right]=-V_{\{f, g\}} .$$

  If $F: M \rightarrow \mathbf{R}$ is a first integral of $(M, H)$, show that the Hamiltonian vector field $V_{F}$ generates a Lie point symmetry of $(\star)$. Prove the converse is also true if $(\star)$ has a fixed point, i.e. a solution of the form $\mathbf{x}(t)=\mathbf{x}_{0}$.

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• Paper 2, Section II, A

  #Integrable Systems

  2017

  Let $U$ and $V$ be non-singular $N \times N$ matrices depending on $(x, t, \lambda)$ which are periodic in $x$ with period $2 \pi$. Consider the associated linear problem

  $$\Psi_{x}=U \Psi, \quad \Psi_{t}=V \Psi$$

  for the vector $\Psi=\Psi(x, t ; \lambda)$. On the assumption that these equations are compatible, derive the zero curvature equation for $(U, V)$.

  Let $W=W(x, t, \lambda)$ denote the $N \times N$ matrix satisfying

  $$W_{x}=U W, \quad W(0, t, \lambda)=I_{N}$$

  where $I_{N}$ is the $N \times N$ identity matrix. You should assume $W$ is unique. By considering $\left(W_{t}-V W\right)_{x}$, show that the matrix $w(t, \lambda)=W(2 \pi, t, \lambda)$ satisfies the Lax equation

  $$w_{t}=[v, w], \quad v(t, \lambda) \equiv V(2 \pi, t, \lambda)$$

  Deduce that $\left\{\operatorname{tr}\left(w^{k}\right)\right\}_{k \geqslant 1}$ are first integrals.

  By considering the matrices

  $$\frac{1}{2 \mathrm{i} \lambda}\left[\begin{array}{cc} \cos u & -\mathrm{i} \sin u \\ \mathrm{i} \sin u & -\cos u \end{array}\right], \quad \frac{\mathrm{i}}{2}\left[\begin{array}{cc} 2 \lambda & u_{x} \\ u_{x} & -2 \lambda \end{array}\right]$$

  show that the periodic Sine-Gordon equation $u_{x t}=\sin u$ has infinitely many first integrals. [You need not prove anything about independence.]

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• Paper 3, Section II, A

  #Integrable Systems

  2017

  Let $u=u(x, t)$ be a smooth solution to the $\mathrm{KdV}$ equation

  $$u_{t}+u_{x x x}-6 u u_{x}=0$$

  which decays rapidly as $|x| \rightarrow \infty$ and let $L=-\partial_{x}^{2}+u$ be the associated Schrödinger operator. You may assume $L$ and $A=4 \partial_{x}^{3}-3\left(u \partial_{x}+\partial_{x} u\right)$ constitute a Lax pair for KdV.

  Consider a solution to $L \varphi=k^{2} \varphi$ which has the asymptotic form

  $$\varphi(x, k, t)= \begin{cases}e^{-\mathrm{i} k x}, & \text { as } x \rightarrow-\infty \\ a(k, t) e^{-\mathrm{i} k x}+b(k, t) e^{\mathrm{i} k x}, & \text { as } x \rightarrow+\infty\end{cases}$$

  Find evolution equations for $a$ and $b$. Deduce that $a(k, t)$ is $t$-independent.

  By writing $\varphi$ in the form

  $$\varphi(x, k, t)=\exp \left[-\mathrm{i} k x+\int_{-\infty}^{x} S(y, k, t) \mathrm{d} y\right], \quad S(x, k, t)=\sum_{n=1}^{\infty} \frac{S_{n}(x, t)}{(2 \mathrm{i} k)^{n}}$$

  show that

  $$a(k, t)=\exp \left[\int_{-\infty}^{\infty} S(x, k, t) \mathrm{d} x\right]$$

  Deduce that $\left\{\int_{-\infty}^{\infty} S_{n}(x, t) \mathrm{d} x\right\}_{n=1}^{\infty}$ are first integrals of KdV.

  By writing a differential equation for $S=X+\mathrm{i} Y$ (with $X, Y$ real), show that these first integrals are trivial when $n$ is even.

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• Paper 3, Section II, F

  #Linear Analysis

  2017

  Let $K$ be a non-empty compact Hausdorff space and let $C(K)$ be the space of real-valued continuous functions on $K$.

  (i) State the real version of the Stone-Weierstrass theorem.

  (ii) Let $A$ be a closed subalgebra of $C(K)$. Prove that $f \in A$ and $g \in A$ implies that $m \in A$ where the function $m: K \rightarrow \mathbb{R}$ is defined by $m(x)=\max \{f(x), g(x)\}$. [You may use without proof that $f \in A$ implies $|f| \in A$.]

  (iii) Prove that $K$ is normal and state Urysohn's Lemma.

  (iv) For any $x \in K$, define $\delta_{x} \in C(K)^{*}$ by $\delta_{x}(f)=f(x)$ for $f \in C(K)$. Justifying your answer carefully, find

  $$\inf _{x \neq y}\left\|\delta_{x}-\delta_{y}\right\| .$$

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• Paper 2, Section II, F

  #Linear Analysis

  2017

  (a) Let $X$ be a normed vector space and $Y \subset X$ a closed subspace with $Y \neq X$. Show that $Y$ is nowhere dense in $X$.

  (b) State any version of the Baire Category theorem.

  (c) Let $X$ be an infinite-dimensional Banach space. Show that $X$ cannot have a countable algebraic basis, i.e. there is no countable subset $\left(x_{k}\right)_{k \in \mathbb{N}} \subset X$ such that every $x \in X$ can be written as a finite linear combination of elements of $\left(x_{k}\right)$.

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• Paper 1, Section II, F

  #Linear Analysis

  2017

  Let $X$ be a normed vector space over the real numbers.

  (a) Define the dual space $X^{*}$ of $X$ and prove that $X^{*}$ is a Banach space. [You may use without proof that $X^{*}$ is a vector space.]

  (b) The Hahn-Banach theorem states the following. Let $X$ be a real vector space, and let $p: X \rightarrow \mathbb{R}$ be sublinear, i.e., $p(x+y) \leqslant p(x)+p(y)$ and $p(\lambda x)=\lambda p(x)$ for all $x, y \in X$ and all $\lambda>0$. Let $Y \subset X$ be a linear subspace, and let $g: Y \rightarrow \mathbb{R}$ be linear and satisfy $g(y) \leqslant p(y)$ for all $y \in Y$. Then there exists a linear functional $f: X \rightarrow \mathbb{R}$ such that $f(x) \leqslant p(x)$ for all $x \in X$ and $\left.f\right|_{Y}=g$.

  Using the Hahn-Banach theorem, prove that for any non-zero $x_{0} \in X$ there exists $f \in X^{*}$ such that $f\left(x_{0}\right)=\left\|x_{0}\right\|$ and $\|f\|=1$.

  (c) Show that $X$ can be embedded isometrically into a Banach space, i.e. find a Banach space $Y$ and a linear map $\Phi: X \rightarrow Y$ with $\|\Phi(x)\|=\|x\|$ for all $x \in X$.

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• Paper 4, Section II, F

  #Linear Analysis

  2017

  Let $H$ be a complex Hilbert space with inner product $(\cdot, \cdot)$ and let $T: H \rightarrow H$ be a bounded linear map.

  (i) Define the spectrum $\sigma(T)$, the point spectrum $\sigma_{p}(T)$, the continuous spectrum $\sigma_{c}(T)$, and the residual spectrum $\sigma_{r}(T)$.

  (ii) Show that $T^{*} T$ is self-adjoint and that $\sigma\left(T^{*} T\right) \subset[0, \infty)$. Show that if $T$ is compact then so is $T^{*} T$.

  (iii) Assume that $T$ is compact. Prove that $T$ has a singular value decomposition: for $N<\infty$ or $N=\infty$, there exist orthonormal systems $\left(u_{i}\right)_{i=1}^{N} \subset H$ and $\left(v_{i}\right)_{i=1}^{N} \subset H$ and $\left(\lambda_{i}\right)_{i=1}^{N} \subset[0, \infty)$ such that, for any $x \in H$,

  $$T x=\sum_{i=1}^{N} \lambda_{i}\left(u_{i}, x\right) v_{i}$$

  [You may use the spectral theorem for compact self-adjoint linear operators.]

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• Paper 3, Section II, H

  #Logic and Set Theory

  2017

  State and prove Zorn's Lemma. [You may assume Hartogs' Lemma.] Indicate clearly where in your proof you have made use of the Axiom of Choice.

  Show that $\mathbb{R}$ has a basis as a vector space over $\mathbb{Q}$.

  Let $V$ be a vector space over $\mathbb{Q}$. Show that all bases of $V$ have the same cardinality.

  [Hint: How does the cardinality of $V$ relate to the cardinality of a given basis?]

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• Paper 2, Section II, H

  #Logic and Set Theory

  2017

  Give the inductive and synthetic definitions of ordinal addition, and prove that they are equivalent.

  Which of the following are always true for ordinals $\alpha, \beta$ and $\gamma$ and which can be false? Give proofs or counterexamples as appropriate.

  (i) $\alpha+\beta=\beta+\alpha$

  (ii) $(\alpha+\beta) \gamma=\alpha \gamma+\beta \gamma$

  (iii) $\alpha(\beta+\gamma)=\alpha \beta+\alpha \gamma$

  (iv) If $\alpha \beta=\beta \alpha$ then $\alpha^{2} \beta^{2}=\beta^{2} \alpha^{2}$

  (v) If $\alpha^{2} \beta^{2}=\beta^{2} \alpha^{2}$ then $\alpha \beta=\beta \alpha$

  [In parts (iv) and (v) you may assume without proof that ordinal multiplication is associative.]

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• Paper 4, Section II, H

  #Logic and Set Theory

  2017

  Prove that every set has a transitive closure. [If you apply the Axiom of Replacement to a function-class $F$, you must explain clearly why $F$ is indeed a function-class.]

  State the Axiom of Foundation and the Principle of $\epsilon$-Induction, and show that they are equivalent (in the presence of the other axioms of $\mathrm{ZFC}$ ).

  State the $\epsilon$-Recursion Theorem.

  Sets $C_{\alpha}$ are defined for each ordinal $\alpha$ by recursion, as follows: $C_{0}=\emptyset, C_{\alpha+1}$ is the set of all countable subsets of $C_{\alpha}$, and $C_{\lambda}=\cup_{\alpha<\lambda} C_{\alpha}$ for $\lambda$ a non-zero limit. Does there exist an $\alpha$ with $C_{\alpha+1}=C_{\alpha}$ ? Justify your answer.

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• Paper 1, Section II, H

  #Logic and Set Theory

  2017

  State the Completeness Theorem for Propositional Logic.

  [You do not need to give definitions of the various terms involved.]

  State the Compactness Theorem and the Decidability Theorem, and deduce them from the Completeness Theorem.

  A set $S$ of propositions is called finitary if there exists a finite set $T$ of propositions such that $\{t: S \vdash t\}=\{t: T \vdash t\}$. Give examples to show that an infinite set of propositions may or may not be finitary.

  Now let $A$ and $B$ be sets of propositions such that every valuation is a model of exactly one of $A$ and $B$. Show that there exist finite subsets $A^{\prime}$ of $A$ and $B^{\prime}$ of $B$ with $A^{\prime} \cup B^{\prime} \models \perp$, and deduce that $A$ and $B$ are finitary.

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• Paper 1, Section I, B

  #Mathematical Biology

  2017

  A model of insect dispersal and growth in one spatial dimension is given by

  $$\frac{\partial N}{\partial t}=D \frac{\partial}{\partial x}\left(N^{2} \frac{\partial N}{\partial x}\right)+\alpha N, \quad N(x, 0)=N_{0} \delta(x),$$

  where $\alpha, D$ and $N_{0}$ are constants, $D>0$, and $\alpha$ may be positive or negative.

  By setting $N(x, t)=R(x, \tau) e^{\alpha t}$, where $\tau(t)$ is some time-like variable satisfying $\tau(0)=0$, show that a suitable choice of $\tau$ yields

  $$R_{\tau}=\left(R^{2} R_{x}\right)_{x}, \quad R(x, 0)=N_{0} \delta(x)$$

  where subscript denotes differentiation with respect to $x$ or $\tau$.

  Consider a similarity solution of the form $R(x, \tau)=F(\xi) / \tau^{\frac{1}{4}}$ where $\xi=x / \tau^{\frac{1}{4}}$. Show that $F$ must satisfy

  $$-\frac{1}{4}(F \xi)^{\prime}=\left(F^{2} F^{\prime}\right)^{\prime} \quad \text { and } \quad \int_{-\infty}^{+\infty} F(\xi) d \xi=N_{0}$$

  [You may use the fact that these are solved by

  $$F(\xi)= \begin{cases}\frac{1}{2} \sqrt{\xi_{0}^{2}-\xi^{2}} & \text { for }|\xi|<\xi_{0} \\ 0 & \text { otherwise }\end{cases}$$

  where $\left.\xi_{0}=\sqrt{4 N_{0} / \pi} .\right]$

  For $\alpha<0$, what is the maximum distance from the origin that insects ever reach? Give your answer in terms of $D, \alpha$ and $N_{0}$.

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• Paper 2, Section I, B

  #Mathematical Biology

  2017

  A bacterial nutrient uptake model is represented by the reaction system

  $$\begin{array}{rll} 2 S+E & \stackrel{k_{1}}{\longrightarrow} & C \\ C & \stackrel{k_{2}}{\longrightarrow} 2 S+E \\ C \stackrel{k_{3}}{\longrightarrow} & E+2 P \end{array}$$

  where the $k_{i}$ are rate constants. Let $s, e, c$ and $p$ represent the concentrations of $S, E, C$ and $P$ respectively. Initially $s=s_{0}, e=e_{0}, c=0$ and $p=0$. Write down the governing differential equation system for the concentrations.

  Either by using the differential equations or directly from the reaction system above, find two invariant quantities. Use these to simplify the system to

  $$\begin{aligned} \dot{s} &=-2 k_{1} s^{2}\left(e_{0}-c\right)+2 k_{2} c \\ \dot{c} &=k_{1} s^{2}\left(e_{0}-c\right)-\left(k_{2}+k_{3}\right) c . \end{aligned}$$

  By setting $u=s / s_{0}$ and $v=c / e_{0}$ and rescaling time, show that the system can be written as

  $$\begin{aligned} u^{\prime} &=-2 u^{2}(1-v)+2(\mu-\lambda) v \\ \epsilon v^{\prime} &=\quad u^{2}(1-v)-\mu v \end{aligned}$$

  where $\epsilon=e_{0} / s_{0}$ and $\mu$ and $\lambda$ should be given. Give the initial conditions for $u$ and $v$.

  [Hint: Note that $2 X$ is equivalent to $X+X$ in reaction systems.]

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• Paper 3, Section I, B

  #Mathematical Biology

  2017

  A stochastic birth-death process has a master equation given by

  $$\frac{d p(n, t)}{d t}=\lambda[p(n-1, t)-p(n, t)]+\beta[(n+1) p(n+1, t)-n p(n, t)]$$

  where $p(n, t)$ is the probability that there are $n$ individuals in the population at time $t$ for $n=0,1,2, \ldots$ and $p(n, t)=0$ for $n<0$.

  Give the corresponding Fokker-Planck equation for this system.

  Use this Fokker-Planck equation to find expressions for $\frac{d}{d t}\langle x\rangle$ and $\frac{d}{d t}\left\langle x^{2}\right\rangle$.

  [Hint: The general form for a Fokker-Planck equation in $P(x, t)$ is

  $$\frac{\partial P}{\partial t}=-\frac{\partial}{\partial x}(A P)+\frac{1}{2} \frac{\partial^{2}}{\partial x^{2}}(B P)$$

  You may use this general form, stating how $A(x)$ and $B(x)$ are constructed. Alternatively, you may derive a Fokker-Plank equation directly by working from the master equation.]

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• Paper 4, Section I, B

  #Mathematical Biology

  2017

  Consider an epidemic model with host demographics (natural births and deaths).

  The system is given by

  $$\begin{aligned} &\frac{d S}{d t}=-\beta I S-\mu S+\mu N \\ &\frac{d I}{d t}=+\beta I S-\nu I-\mu I \end{aligned}$$

  where $S(t)$ are the susceptibles, $I(t)$ are the infecteds, $N$ is the total population size and the parameters $\beta, \mu$ and $\nu$ are positive. The basic reproduction ratio is defined as $R_{0}=\beta N /(\mu+\nu) .$

  Show that the system has an endemic equilibrium (where the disease is present) for $R_{0}>1$. Show that the endemic equilibrium is stable.

  Interpret the meaning of the case $\nu \gg \mu$ and show that in this case the approximate period of (decaying) oscillation around the endemic equilibrium is given by

  $$T=\frac{2 \pi}{\sqrt{\mu \nu\left(R_{0}-1\right)}}$$

  Suppose now a vaccine is introduced which is given to some proportion of the population at birth, but not enough to eradicate the disease. What will be the effect on the period of (decaying) oscillations?

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• Paper 3, Section II, B

  #Mathematical Biology

  2017

  In a discrete-time model, adults and larvae of a population at time $n$ are represented by $a_{n}$ and $b_{n}$ respectively. The model is represented by the equations

  $$\begin{aligned} a_{n+1} &=(1-k) a_{n}+\frac{b_{n}}{1+a_{n}} \\ b_{n+1} &=\mu a_{n} \end{aligned}$$

  You may assume that $k \in(0,1)$ and $\mu>0$. Give an explanation of what each of the terms represents, and hence give a description of the population model.

  By combining the equations to describe the dynamics purely in terms of the adults, find all equilibria of the system. Show that the equilibrium with the population absent $(a=0)$ is unstable exactly when there exists an equilibrium with the population present $(a>0)$.

  Give the condition on $\mu$ and $k$ for the equilibrium with $a>0$ to be stable, and sketch the corresponding region in the $(k, \mu)$ plane.

  What happens to the population close to the boundaries of this region?

  If this model was modified to include stochastic effects, briefly describe qualitatively the likely dynamics near the boundaries of the region found above.

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• Paper 4, Section II, B

  #Mathematical Biology

  2017

  An activator-inhibitor system is described by the equations

  $$\begin{aligned} &\frac{\partial u}{\partial t}=u(c+u-v)+\frac{\partial^{2} u}{\partial x^{2}} \\ &\frac{\partial v}{\partial t}=v(a u-b v)+d \frac{\partial^{2} v}{\partial x^{2}} \end{aligned}$$

  where $a, b, c, d>0$.

  Find and sketch the range of $a, b$ for which the spatially homogeneous system has a stable stationary solution with $u>0$ and $v>0$.

  Considering spatial perturbations of the form $\cos (k x)$ about the solution found above, find conditions for the system to be unstable. Sketch this region in the $(a, b)$-plane for fixed $d$ (for a value of $d$ such that the region is non-empty).

  Show that $k_{c}$, the critical wavenumber at the onset of the instability, is given by

  $$k_{c}=\sqrt{\frac{2 a c}{d-a}}$$

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• Paper 2, Section II, 18H

  #Number Fields

  2017

  (a) Let $L$ be a number field, $\mathcal{O}_{L}$ the ring of integers in $L, \mathcal{O}_{L}^{*}$ the units in $\mathcal{O}_{L}, r$ the number of real embeddings of $L$, and $s$ the number of pairs of complex embeddings of $L$.

  Define a group homomorphism $\mathcal{O}_{L}^{*} \rightarrow \mathbb{R}^{r+s-1}$ with finite kernel, and prove that the image is a discrete subgroup of $\mathbb{R}^{r+s-1}$.

  (b) Let $K=\mathbb{Q}(\sqrt{d})$ where $d>1$ is a square-free integer. What is the structure of the group of units of $K$ ? Show that if $d$ is divisible by a prime $p \equiv 3(\bmod 4)$ then every unit of $K$ has norm $+1$. Find an example of $K$ with a unit of norm $-1$.

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• Paper 1, Section II, H

  #Number Fields

  2017

  Let $\mathcal{O}_{L}$ be the ring of integers in a number field $L$, and let $\mathfrak{a} \leqslant \mathcal{O}_{L}$ be a non-zero ideal of $\mathcal{O}_{L}$.

  (a) Show that $\mathfrak{a} \cap \mathbb{Z} \neq\{0\}$.

  (b) Show that $\mathcal{O}_{L} / \mathfrak{a}$ is a finite abelian group.

  (c) Show that if $x \in L$ has $x \mathfrak{a} \subseteq \mathfrak{a}$, then $x \in \mathcal{O}_{L}$.

  (d) Suppose $[L: \mathbb{Q}]=2$, and $\mathfrak{a}=\langle b, \alpha\rangle$, with $b \in \mathbb{Z}$ and $\alpha \in \mathcal{O}_{L}$. Show that $\langle b, \alpha\rangle\langle b, \bar{\alpha}\rangle$ is principal.

  [You may assume that $\mathfrak{a}$ has an integral basis.]

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• Paper 4, Section II, H

  #Number Fields

  2017

  (a) Write down $\mathcal{O}_{K}$, when $K=\mathbb{Q}(\sqrt{\delta})$, and $\delta \equiv 2$ or $3(\bmod 4)$. [You need not prove your answer.]

  Let $L=\mathbb{Q}(\sqrt{2}, \sqrt{\delta})$, where $\delta \equiv 3(\bmod 4)$ is a square-free integer. Find an integral basis of $\mathcal{O}_{L} \cdot$ [Hint: Begin by considering the relative traces $t r_{L / K}$, for $K$ a quadratic subfield of $L .]$

  (b) Compute the ideal class group of $\mathbb{Q}(\sqrt{-14})$.

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• Paper 1, Section $I$, G

  #Number Theory

  2017

  Define the Legendre symbol $\left(\frac{a}{p}\right)$.

  State Gauss' lemma and use it to compute $\left(\frac{2}{p}\right)$ where $p$ is an odd prime.

  Show that if $m \geqslant 4$ is a power of 2 , and $p$ is a prime dividing $2^{m}+1$, then $p \equiv 1(\bmod 4 m)$.

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• Paper 3, Section I, G

  #Number Theory

  2017

  Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler pseudoprime to the base $b$ if and only if $b^{2} \equiv \pm 1(\bmod 65)$. How many such bases are there? Show that the bases for which 65 is a strong pseudoprime do not form a subgroup of $(\mathbb{Z} / 65 \mathbb{Z})^{\times}$.

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• Paper 4, Section I, G

  #Number Theory

  2017

  Show that, for $x \geqslant 2$ a real number,

  $$\prod_{\substack{p \leqslant x \\ p \text { is prime }}}\left(1-\frac{1}{p}\right)^{-1}>\log x$$

  Hence prove that

  $$\sum_{\substack{p \leqslant x, p \text { is prime }}} \frac{1}{p}>\log \log x+c,$$

  where $c$ is a constant you should make explicit.

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• Paper 2, Section I, G

  #Number Theory

  2017

  State and prove Legendre's formula for $\pi(x)$. Use it to compute $\pi(42)$.

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• Paper 3, Section II, G

  #Number Theory

  2017

  Let $d$ be a positive integer which is not a square. Assume that the continued fraction expansion of $\sqrt{d}$ takes the form $\left[a_{0}, \overline{a_{1}, a_{2}, \ldots, a_{m}}\right]$.

  (a) Define the convergents $p_{n} / q_{n}$, and show that $p_{n}$ and $q_{n}$ are coprime.

  (b) The complete quotients $\theta_{n}$ may be written in the form $\left(\sqrt{d}+r_{n}\right) / s_{n}$, where $r_{n}$ and $s_{n}$ are rational numbers. Use the relation

  $$\sqrt{d}=\frac{\theta_{n} p_{n-1}+p_{n-2}}{\theta_{n} q_{n-1}+q_{n-2}}$$

  to find formulae for $r_{n}$ and $s_{n}$ in terms of the $p$ 's and $q$ 's. Deduce that $r_{n}$ and $s_{n}$ are integers.

  (c) Prove that Pell's equation $x^{2}-d y^{2}=1$ has infinitely many solutions in integers $x$ and $y$.

  (d) Find integers $x$ and $y$ satisfying $x^{2}-67 y^{2}=-2$.

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• Paper 4, Section II, 10G

  #Number Theory

  2017

  (a) State Dirichlet's theorem on primes in arithmetic progression.

  (b) Let $d$ be the discriminant of a binary quadratic form, and let $p$ be an odd prime. Show that $p$ is represented by some binary quadratic form of discriminant $d$ if and only if $x^{2} \equiv d(\bmod p)$ is soluble.

  (c) Let $f(x, y)=x^{2}+15 y^{2}$ and $g(x, y)=3 x^{2}+5 y^{2}$. Show that $f$ and $g$ each represent infinitely many primes. Are there any primes represented by both $f$ and $g$ ?

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• Paper 2, Section II, A

  #Numerical Analysis

  2017

  The Poisson equation $\nabla^{2} u=f$ in the unit square $\Omega=[0,1] \times[0,1]$, equipped with the zero Dirichlet boundary conditions on $\partial \Omega$, is discretized with the nine-point formula:

  $$\begin{aligned} \Gamma_{9}\left[u_{i, j}\right]:=&-\frac{10}{3} u_{i, j}+\frac{2}{3}\left(u_{i+1, j}+u_{i-1, j}+u_{i, j+1}+u_{i, j-1}\right) \\ &+\frac{1}{6}\left(u_{i+1, j+1}+u_{i+1, j-1}+u_{i-1, j+1}+u_{i-1, j-1}\right)=h^{2} f_{i, j} \end{aligned}$$

  where $1 \leqslant i, j \leqslant m, u_{i, j} \approx u(i h, j h)$, and $(i h, j h)$ are the grid points with $h=\frac{1}{m+1}$.

  (i) Find the order of the local truncation error $\eta_{i, j}$ of the approximation.

  (ii) Prove that the order of the truncation error is smaller if $f$ satisfies the Laplace equation $\nabla^{2} f=0$.

  (iii) Show that the modified nine-point scheme

  $$\begin{aligned} \Gamma_{9}\left[u_{i, j}\right] &=h^{2} f_{i, j}+\frac{1}{12} h^{2} \Gamma_{5}\left[f_{i, j}\right] \\ &:=h^{2} f_{i, j}+\frac{1}{12} h^{2}\left(f_{i+1, j}+f_{i-1, j}+f_{i, j+1}+f_{i, j-1}-4 f_{i, j}\right) \end{aligned}$$

  has a truncation error of the same order as in part (ii).

  (iv) Let $\left(u_{i, j}\right)_{i, j=1}^{m}$ be a solution to the $m^{2} \times m^{2}$ system of linear equations $A \mathbf{u}=\mathbf{b}$ arising from the modified nine-point scheme in part (iii). Further, let $u(x, y)$ be the exact solution and let $e_{i, j}:=u_{i, j}-u(i h, j h)$ be the error of approximation at grid points. Prove that there exists a constant $c$ such that

  $$\|\mathbf{e}\|_{2}:=\left[\sum_{i, j=1}^{m}\left|e_{i, j}\right|^{2}\right]^{1 / 2}<c h^{3}, \quad h \rightarrow 0$$

  [Hint: The nine-point discretization of $\nabla^{2} u$ can be written as

  $$\Gamma_{9}[u]=\left(\Gamma_{5}+\frac{1}{6} \Delta_{x}^{2} \Delta_{y}^{2}\right) u=\left(\Delta_{x}^{2}+\Delta_{y}^{2}+\frac{1}{6} \Delta_{x}^{2} \Delta_{y}^{2}\right) u$$

  where $\Gamma_{5}[u]=\left(\Delta_{x}^{2}+\Delta_{y}^{2}\right) u$ is the five-point discretization and

  $$\left.\begin{array}{l} \Delta_{x}^{2} u(x, y):=u(x-h, y)-2 u(x, y)+u(x+h, y) \\ \Delta_{y}^{2} u(x, y):=u(x, y-h)-2 u(x, y)+u(x, y+h) \end{array}\right]$$

  [Hint: The matrix A of the nine-point scheme is symmetric, with the eigenvalues

  $$\lambda_{k, \ell}=-4 \sin ^{2} \frac{k \pi h}{2}-4 \sin ^{2} \frac{\ell \pi h}{2}+\frac{8}{3} \sin ^{2} \frac{k \pi h}{2} \sin ^{2} \frac{\ell \pi h}{2}, \quad 1 \leqslant k, \ell \leqslant m$$

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• Paper 1, Section II, A

  #Numerical Analysis

  2017

  State the Householder-John theorem and explain how it can be used in designing iterative methods for solving a system of linear equations $A \mathbf{x}=\mathbf{b}$. [You may quote other relevant theorems if needed.]

  Consider the following iterative scheme for solving $A \mathbf{x}=\mathbf{b}$. Let $A=L+D+U$, where $D$ is the diagonal part of $A$, and $L$ and $U$ are the strictly lower and upper triangular parts of $A$, respectively. Then, with some starting vector $\mathbf{x}^{(0)}$, the scheme is as follows:

  $$(D+\omega L) \mathbf{x}^{(k+1)}=[(1-\omega) D-\omega U] \mathbf{x}^{(k)}+\omega \mathbf{b} .$$

  Prove that if $A$ is a symmetric positive definite matrix and $\omega \in(0,2)$, then, for any $\mathbf{x}^{(0)}$, the above iteration converges to the solution of the system $A \mathbf{x}=\mathbf{b}$.

  Which method corresponds to the case $\omega=1 ?$

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• Paper 3, Section II, A

  #Numerical Analysis

  2017

  Let $A$ be a real symmetric $n \times n$ matrix with real and distinct eigenvalues $0=\lambda_{1}<\cdots<\lambda_{n-1}=1<\lambda_{n}$ and a corresponding orthogonal basis of normalized real eigenvectors $\left(\mathbf{w}_{i}\right)_{i=1}^{n}$.

  To estimate the eigenvector $\mathbf{w}_{n}$ of $A$ whose eigenvalue is $\lambda_{n}$, the power method with shifts is employed which has the following form:

  $$\mathbf{y}=\left(A-s_{k} I\right) \mathbf{x}^{(k)}, \quad \mathbf{x}^{(k+1)}=\mathbf{y} /\|\mathbf{y}\|, \quad s_{k} \in \mathbb{R}, \quad k=0,1,2, \ldots$$

  Three versions of this method are considered:

  (i) no shift: $s_{k} \equiv 0$;

  (ii) single shift: $s_{k} \equiv \frac{1}{2}$;

  (iii) double shift: $s_{2 \ell} \equiv s_{0}=\frac{1}{4}(2+\sqrt{2}), s_{2 \ell+1} \equiv s_{1}=\frac{1}{4}(2-\sqrt{2})$.

  Assume that $\lambda_{n}=1+\epsilon$, where $\epsilon>0$ is very small, so that the terms $\mathcal{O}\left(\epsilon^{2}\right)$ are negligible, and that $\mathbf{x}^{(0)}$ contains substantial components of all the eigenvectors.

  By considering the approximation after $2 m$ iterations in the form

  $$\mathbf{x}^{(2 m)}=\pm \mathbf{w}_{n}+\mathcal{O}\left(\rho^{2 m}\right) \quad(m \rightarrow \infty)$$

  find $\rho$ as a function of $\epsilon$ for each of the three versions of the method.

  Compare the convergence rates of the three versions of the method, with reference to the number of iterations needed to achieve a prescribed accuracy.

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• Paper 4, Section II, A

  #Numerical Analysis

  2017

  (a) The diffusion equation

  $$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}, \quad 0 \leqslant x \leqslant 1, \quad 0 \leqslant t \leqslant T$$

  is approximated by the Crank-Nicolson scheme

  $$u_{m}^{n+1}-\frac{1}{2} \mu\left(u_{m-1}^{n+1}-2 u_{m}^{n+1}+u_{m+1}^{n+1}\right)=u_{m}^{n}+\frac{1}{2} \mu\left(u_{m-1}^{n}-2 u_{m}^{n}+u_{m+1}^{n}\right)$$

  with $m=1, \ldots, M$. Here $\mu=k / h^{2}, k=\Delta t, h=\Delta x=\frac{1}{M+1}$, and $u_{m}^{n}$ is an approximation to $u(m h, n k)$. Assuming that $u(0, t)=u(1, t) \stackrel{M+1}{=} 0$, show that the above scheme can be written in the form

  $$B \mathbf{u}^{n+1}=C \mathbf{u}^{n}, \quad 0 \leqslant n \leqslant T / k-1$$

  where $\mathbf{u}^{n}=\left[u_{1}^{n}, \ldots, u_{M}^{n}\right]^{T}$ and the real matrices $B$ and $C$ should be found. Using matrix analysis, find the range of $\mu>0$ for which the scheme is stable.

  [Hint: All Toeplitz symmetric tridiagonal (TST) matrices have the same set of orthogonal eigenvectors, and a TST matrix with the elements $a_{i, i}=a$ and $a_{i, i \pm 1}=b$ has the eigenvalues $\lambda_{k}=a+2 b \cos \frac{\pi k}{M+1}$. ]

  (b) The wave equation

  $$\frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial^{2} u}{\partial x^{2}}, \quad x \in \mathbb{R}, \quad t \geqslant 0$$

  with given initial conditions for $u$ and $\partial u / \partial t$, is approximated by the scheme

  $$u_{m}^{n+1}-2 u_{m}^{n}+u_{m}^{n-1}=\mu\left(u_{m+1}^{n}-2 u_{m}^{n}+u_{m-1}^{n}\right),$$

  with the Courant number now $\mu=k^{2} / h^{2}$. Applying the Fourier technique, find the range of $\mu>0$ for which the method is stable.

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• Paper 2, Section II, K

  #Optimization and Control

  2017

  During each of $N$ time periods a venture capitalist, Vicky, is presented with an investment opportunity for which the rate of return for that period is a random variable; the rates of return in successive periods are independent identically distributed random variables with distributions concentrated on $[-1, \infty)$. Thus, if $x_{n}$ is Vicky's capital at period $n$, then $x_{n+1}=\left(1-p_{n}\right) x_{n}+p_{n} x_{n}\left(1+R_{n}\right)$, where $p_{n} \in[0,1]$ is the proportion of her capital she chooses to invest at period $n$, and $R_{n}$ is the rate of return for period $n$. Vicky desires to maximize her expected yield over $N$ periods, where the yield is defined as $\left(\frac{x_{N}}{x_{0}}\right)^{\frac{1}{N}}-1$, and $x_{0}$ and $x_{N}$ are respectively her initial and final capital.

  (a) Express the problem of finding an optimal policy in a dynamic programming framework.

  (b) Show that in each time period, the optimal strategy can be expressed in terms of the quantity $p^{*}$ which solves the optimization problem $\max _{p} \mathbb{E}\left(1+p R_{1}\right)^{1 / N}$. Show that $p^{*}>0$ if $\mathbb{E} R_{1}>0$. [Do not calculate $p^{*}$ explicitly.]

  (c) Compare her optimal policy with the policy which maximizes her expected final capital $x_{N}$.

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• Paper 3, Section II, K

  #Optimization and Control

  2017

  A particle follows a discrete-time trajectory on $\mathbb{R}$ given by

  $$x_{t+1}=\left(A x_{t}+u_{t}\right) \xi_{t}+\epsilon_{t}$$

  for $t=1,2, \ldots, T$. Here $T \geqslant 2$ is a fixed integer, $A$ is a real constant, $x_{t}$ and $u_{t}$ are the position of the particle and control action at time $t$, respectively, and $\left(\xi_{t}, \epsilon_{t}\right)_{t=1}^{T}$ is a sequence of independent random vectors with

  $$\mathbb{E} \xi_{t}=\mathbb{E} \epsilon_{t}=0, \operatorname{var}\left(\xi_{t}\right)=V_{\xi}>0, \operatorname{var}\left(\epsilon_{t}\right)=V_{\epsilon}>0 \text { and } \operatorname{cov}\left(\xi_{t}, \epsilon_{t}\right)=0$$

  Find the optimal control, i.e. the control action $u_{t}$, defined as a function of $\left(x_{1}, \ldots, x_{t} ; u_{1}, \ldots, u_{t-1}\right)$, that minimizes

  $$\sum_{t=1}^{T} x_{t}^{2}+c \sum_{t=1}^{T-1} u_{t}^{2}$$

  where $c>0$ is given.

  On which of $V_{\epsilon}$ and $V_{\xi}$ does the optimal control depend?

  Find the limiting form of the optimal control as $T \rightarrow \infty$, and the minimal average cost per unit time.

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• Paper 4, Section II, $29 \mathrm{~K}$

  #Optimization and Control

  2017

  A file of $X$ gigabytes (GB) is to be transmitted over a communications link. At each time $t$ the sender can choose a transmission rate $u(t)$ within the range $[0,1]$ GB per second. The charge for transmitting at rate $u(t)$ at time $t$ is $u(t) p(t)$. The function $p$ is fully known at time $t=0$. If it takes a total time $T$ to transmit the file then there is a delay cost of $\gamma T^{2}, \gamma>0$. Thus $u$ and $T$ are to be chosen to minimize

  $$\int_{0}^{T} u(t) p(t) d t+\gamma T^{2}$$

  where $u(t) \in[0,1], d x(t) / d t=-u(t), x(0)=X$ and $x(T)=0$. Using Pontryagin's maximum principle, or otherwise, show that a property of the optimal policy is that there exists $p^{*}$ such that $u(t)=1$ if $p(t)<p^{*}$ and $u(t)=0$ if $p(t)>p^{*}$.

  Show that the optimal $p^{*}$ and $T$ are related by $p^{*}=p(T)+2 \gamma T$.

  Suppose $p(t)=t+1 / t$ and $X=1$. Show that it is optimal to transmit at a constant rate $u(t)=1$ between times $T-1 \leqslant t \leqslant T$, where $T$ is the unique positive solution to the equation

  $$\frac{1}{(T-1) T}=2 \gamma T+1$$

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• Paper 1, Section II, C

  #Principles of Quantum Mechanics

  2017

  The position and momentum operators of the harmonic oscillator can be written as

  $$\hat{x}=\left(\frac{\hbar}{2 m \omega}\right)^{1 / 2}\left(a+a^{\dagger}\right), \quad \hat{p}=\left(\frac{\hbar m \omega}{2}\right)^{1 / 2} i\left(a^{\dagger}-a\right)$$

  where $m$ is the mass, $\omega$ is the frequency and the Hamiltonian is

  $$H=\frac{1}{2 m} \hat{p}^{2}+\frac{1}{2} m \omega^{2} \hat{x}^{2}$$

  Assuming that

  $$[\hat{x}, \hat{p}]=i \hbar$$

  derive the commutation relations for $a$ and $a^{\dagger}$. Construct the Hamiltonian in terms of $a$ and $a^{\dagger}$. Assuming that there is a unique ground state, explain how all other energy eigenstates can be constructed from it. Determine the energy of each of these eigenstates.

  Consider the modified Hamiltonian

  $$H^{\prime}=H+\lambda \hbar \omega\left(a^{2}+a^{\dagger 2}\right)$$

  where $\lambda$ is a dimensionless parameter. Use perturbation theory to calculate the modified energy levels to second order in $\lambda$, quoting any standard formulae that you require. Show that the modified Hamiltonian can be written as

  $$H^{\prime}=\frac{1}{2 m}(1-2 \lambda) \hat{p}^{2}+\frac{1}{2} m \omega^{2}(1+2 \lambda) \hat{x}^{2} .$$

  Assuming $|\lambda|<\frac{1}{2}$, calculate the modified energies exactly. Show that the results are compatible with those obtained from perturbation theory.

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• Paper 2, Section II, C

  #Principles of Quantum Mechanics

  2017

  Let $\boldsymbol{\sigma}=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$ be a set of Hermitian operators obeying

  $$\left[\sigma_{i}, \sigma_{j}\right]=2 i \epsilon_{i j k} \sigma_{k} \quad \text { and } \quad(\mathbf{n} \cdot \boldsymbol{\sigma})^{2}=1$$

  where $\mathbf{n}$ is any unit vector. Show that $(*)$ implies that

  $$(\mathbf{a} \cdot \boldsymbol{\sigma})(\mathbf{b} \cdot \boldsymbol{\sigma})=\mathbf{a} \cdot \mathbf{b}+i(\mathbf{a} \times \mathbf{b}) \cdot \boldsymbol{\sigma}$$

  for any vectors a and $\mathbf{b}$. Explain, with reference to the properties $(*)$, how $\boldsymbol{\sigma}$ can be related to the intrinsic angular momentum $\mathbf{S}$ for a particle of spin $\frac{1}{2}$.

  Show that the operators $P_{\pm}=\frac{1}{2}(1 \pm \mathbf{n} \cdot \boldsymbol{\sigma})$ are Hermitian and obey

  $$P_{\pm}^{2}=P_{\pm}, \quad P_{+} P_{-}=P_{-} P_{+}=0$$

  Show how $P_{\pm}$can be used to write any state $|\chi\rangle$ as a linear combination of eigenstates of $\mathbf{n} \cdot \boldsymbol{\sigma}$. Use this to deduce that if the system is in a normalised state $|\chi\rangle$ when $\mathbf{n} \cdot \boldsymbol{\sigma}$ is measured, then the results $\pm 1$ will be obtained with probabilities

  $$\| P_{\pm}|\chi\rangle \|^{2}=\frac{1}{2}(1 \pm\langle\chi|\mathbf{n} \cdot \boldsymbol{\sigma}| \chi\rangle)$$

  If $|\chi\rangle$ is a state corresponding to the system having spin up along a direction defined by a unit vector $\mathbf{m}$, show that a measurement will find the system to have spin up along $\mathbf{n}$ with probability $\frac{1}{2}(1+\mathbf{n} \cdot \mathbf{m})$.

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• Paper 3, Section II, C

  #Principles of Quantum Mechanics

  2017

  The angular momentum operators $\mathbf{J}=\left(J_{1}, J_{2}, J_{3}\right)$ obey the commutation relations

  $$\begin{aligned} &{\left[J_{3}, J_{\pm}\right]=\pm J_{\pm}} \\ &{\left[J_{+}, J_{-}\right]=2 J_{3}} \end{aligned}$$

  where $J_{\pm}=J_{1} \pm i J_{2}$.

  A quantum mechanical system involves the operators $a, a^{\dagger}, b$ and $b^{\dagger}$ such that

  $$\begin{gathered} {\left[a, a^{\dagger}\right]=\left[b, b^{\dagger}\right]=1} \\ {[a, b]=\left[a^{\dagger}, b\right]=\left[a, b^{\dagger}\right]=\left[a^{\dagger}, b^{\dagger}\right]=0 .} \end{gathered}$$

  Define $K_{+}=a^{\dagger} b, K_{-}=a b^{\dagger}$ and $K_{3}=\frac{1}{2}\left(a^{\dagger} a-b^{\dagger} b\right)$. Show that $K_{\pm}$and $K_{3}$ obey the same commutation relations as $J_{\pm}$and $J_{3}$.

  Suppose that the system is in the state $|0\rangle$ such that $a|0\rangle=b|0\rangle=0$. Show that $\left(a^{\dagger}\right)^{2}|0\rangle$ is an eigenstate of $K_{3}$. Let $K^{2}=\frac{1}{2}\left(K_{+} K_{-}+K_{-} K_{+}\right)+K_{3}^{2}$. Show that $\left(a^{\dagger}\right)^{2}|0\rangle$ is an eigenstate of $K^{2}$ and find the eigenvalue. How many other states do you expect to find with same value of $K^{2}$ ? Find them.

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• Paper 4, Section II, C

  #Principles of Quantum Mechanics

  2017

  The Hamiltonian for a quantum system in the Schrödinger picture is

  $$H_{0}+\lambda V(t),$$

  where $H_{0}$ is independent of time and the parameter $\lambda$ is small. Define the interaction picture corresponding to this Hamiltonian and derive a time evolution equation for interaction picture states.

  Let $|n\rangle$ and $|m\rangle$ be eigenstates of $H_{0}$ with distinct eigenvalues $E_{n}$ and $E_{m}$ respectively. Show that if the system was in the state $|n\rangle$ in the remote past, then the probability of measuring it to be in a different state $|m\rangle$ at a time $t$ is

  $$\frac{\lambda^{2}}{\hbar^{2}}\left|\int_{-\infty}^{t} d t^{\prime}\left\langle m\left|V\left(t^{\prime}\right)\right| n\right\rangle e^{i\left(E_{m}-E_{n}\right) t^{\prime} / \hbar}\right|^{2}+O\left(\lambda^{3}\right)$$

  Let the system be a simple harmonic oscillator with $H_{0}=\hbar \omega\left(a^{\dagger} a+\frac{1}{2}\right)$, where $\left[a, a^{\dagger}\right]=1$. Let $|0\rangle$ be the ground state which obeys $a|0\rangle=0$. Suppose

  $$V(t)=e^{-p|t|}\left(a+a^{\dagger}\right),$$

  with $p>0$. In the remote past the system was in the ground state. Find the probability, to lowest non-trivial order in $\lambda$, for the system to be in the first excited state in the far future.

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• Paper 2, Section II, $26 \mathrm{~K}$

  #Principles of Statistics

  2017

  We consider the problem of estimating $\theta$ in the model $\{f(x, \theta): \theta \in(0, \infty)\}$, where

  $$f(x, \theta)=(1-\alpha)(x-\theta)^{-\alpha} 1\{x \in[\theta, \theta+1]\}$$

  Here $1\{A\}$ is the indicator of the set $A$, and $\alpha \in(0,1)$ is known. This estimation is based on a sample of $n$ i.i.d. $X_{1}, \ldots, X_{n}$, and we denote by $X_{(1)}<\ldots<X_{(n)}$ the ordered sample.

  (a) Compute the mean and the variance of $X_{1}$. Construct an unbiased estimator of $\theta$ taking the form $\tilde{\theta}_{n}=\bar{X}_{n}+c(\alpha)$, where $\bar{X}_{n}=n^{-1} \sum_{i=1}^{n} X_{i}$, specifying $c(\alpha)$.

  (b) Show that $\tilde{\theta}_{n}$ is consistent and find the limit in distribution of $\sqrt{n}\left(\tilde{\theta}_{n}-\theta\right)$. Justify your answer, citing theorems that you use.

  (c) Find the maximum likelihood estimator $\hat{\theta}_{n}$ of $\theta$. Compute $\mathbf{P}\left(\hat{\theta}_{n}-\theta>t\right)$ for all real $t$. Is $\hat{\theta}_{n}$ unbiased?

  (d) For $t>0$, show that $\mathbf{P}\left(n^{\beta}\left(\hat{\theta}_{n}-\theta\right)>t\right)$ has a limit in $(0,1)$ for some $\beta>0$. Give explicitly the value of $\beta$ and the limit. Why should one favour using $\hat{\theta}_{n}$ over $\tilde{\theta}_{n}$ ?

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• Paper 3, Section II, $26 \mathrm{~K}$

  #Principles of Statistics

  2017

  We consider the problem of estimating an unknown $\theta_{0}$ in a statistical model $\{f(x, \theta), \theta \in \Theta\}$ where $\Theta \subset \mathbb{R}$, based on $n$ i.i.d. observations $X_{1}, \ldots, X_{n}$ whose distribution has p.d.f. $f\left(x, \theta_{0}\right)$.

  In all the parts below you may assume that the model satisfies necessary regularity conditions.

  (a) Define the score function $S_{n}$ of $\theta$. Prove that $S_{n}\left(\theta_{0}\right)$ has mean 0 .

  (b) Define the Fisher Information $I(\theta)$. Show that it can also be expressed as

  $$I(\theta)=-\mathbb{E}_{\theta}\left[\frac{d^{2}}{d \theta^{2}} \log f\left(X_{1}, \theta\right)\right]$$

  (c) Define the maximum likelihood estimator $\hat{\theta}_{n}$ of $\theta$. Give without proof the limits of $\hat{\theta}_{n}$ and of $\sqrt{n}\left(\hat{\theta}_{n}-\theta_{0}\right.$ ) (in a manner which you should specify). [Be as precise as possible when describing a distribution.]

  (d) Let $\psi: \Theta \rightarrow \mathbb{R}$ be a continuously differentiable function, and $\tilde{\theta}_{n}$ another estimator of $\theta_{0}$ such that $\left|\hat{\theta}_{n}-\tilde{\theta}_{n}\right| \leqslant 1 / n$ with probability 1 . Give the limits of $\psi\left(\tilde{\theta}_{n}\right)$ and of $\sqrt{n}\left(\psi\left(\tilde{\theta}_{n}\right)-\psi\left(\theta_{0}\right)\right)$ (in a manner which you should specify).

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• Paper 4, Section II, $27 \mathrm{~K}$

  #Principles of Statistics

  2017

  For the statistical model $\left\{\mathcal{N}_{d}(\theta, \Sigma), \theta \in \mathbb{R}^{d}\right\}$, where $\Sigma$ is a known, positive-definite $d \times d$ matrix, we want to estimate $\theta$ based on $n$ i.i.d. observations $X_{1}, \ldots, X_{n}$ with distribution $\mathcal{N}_{d}(\theta, \Sigma)$.

  (a) Derive the maximum likelihood estimator $\hat{\theta}_{n}$ of $\theta$. What is the distribution of $\hat{\theta}_{n}$ ?

  (b) For $\alpha \in(0,1)$, construct a confidence region $C_{n}^{\alpha}$ such that $\mathbf{P}_{\theta}\left(\theta \in C_{n}^{\alpha}\right)=1-\alpha$.

  (c) For $\Sigma=I_{d}$, compute the maximum likelihood estimator of $\theta$ for the following parameter spaces:

  (i) $\Theta=\left\{\theta:\|\theta\|_{2}=1\right\}$.

  (ii) $\Theta=\left\{\theta: v^{\top} \theta=0\right\}$ for some unit vector $v \in \mathbb{R}^{d}$.

  (d) For $\Sigma=I_{d}$, we want to test the null hypothesis $\Theta_{0}=\{0\}$ (i.e. $\left.\theta=0\right)$ against the composite alternative $\Theta_{1}=\mathbb{R}^{d} \backslash\{0\}$. Compute the likelihood ratio statistic $\Lambda\left(\Theta_{1}, \Theta_{0}\right)$ and give its distribution under the null hypothesis. Compare this result with the statement of Wilks' theorem.

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• Paper 1, Section II, $28 \mathrm{~K}$

  #Principles of Statistics

  2017

  For a positive integer $n$, we want to estimate the parameter $p$ in the binomial statistical model $\{\operatorname{Bin}(n, p), p \in[0,1]\}$, based on an observation $X \sim \operatorname{Bin}(n, p)$.

  (a) Compute the maximum likelihood estimator for $p$. Show that the posterior distribution for $p$ under a uniform prior on $[0,1]$ is $\operatorname{Beta}(a, b)$, and specify $a$ and $b$. [The p.d.f. of $\operatorname{Beta}(a, b)$ is given by

  $$\left.f_{a, b}(p)=\frac{(a+b-1) !}{(a-1) !(b-1) !} p^{a-1}(1-p)^{b-1} .\right]$$

  (b) (i) For a risk function $L$, define the risk of an estimator $\hat{p}$ of $p$, and the Bayes risk under a prior $\pi$ for $p$.

  (ii) Under the loss function

  $$L(\hat{p}, p)=\frac{(\hat{p}-p)^{2}}{p(1-p)}$$

  find a Bayes optimal estimator for the uniform prior. Give its risk as a function of $p$.

  (iii) Give a minimax optimal estimator for the loss function $L$ given above. Justify your answer.

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• Paper 2, Section II, J

  #Probability and Measure

  2017

  (a) Give the definition of the Fourier transform $\widehat{f}$ of a function $f \in L^{1}\left(\mathbb{R}^{d}\right)$.

  (b) Explain what it means for Fourier inversion to hold.

  (c) Prove that Fourier inversion holds for $g_{t}(x)=(2 \pi t)^{-d / 2} e^{-\|x\|^{2} /(2 t)}$. Show all of the steps in your computation. Deduce that Fourier inversion holds for Gaussian convolutions, i.e. any function of the form $f * g_{t}$ where $t>0$ and $f \in L^{1}\left(\mathbb{R}^{d}\right)$.

  (d) Prove that any function $f$ for which Fourier inversion holds has a bounded, continuous version. In other words, there exists $g$ bounded and continuous such that $f(x)=g(x)$ for a.e. $x \in \mathbb{R}^{d}$.

  (e) Does Fourier inversion hold for $f=\mathbf{1}_{[0,1]}$ ?

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• Paper 3, Section II, J

  #Probability and Measure

  2017

  (a) Suppose that $\mathcal{X}=\left(X_{n}\right)$ is a sequence of random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Give the definition of what it means for $\mathcal{X}$ to be uniformly integrable.

  (b) State and prove Hölder's inequality.

  (c) Explain what it means for a family of random variables to be $L^{p}$ bounded. Prove that an $L^{p}$ bounded sequence is uniformly integrable provided $p>1$.

  (d) Prove or disprove: every sequence which is $L^{1}$ bounded is uniformly integrable.

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• Paper 4, Section II, J

  #Probability and Measure

  2017

  (a) Suppose that $(E, \mathcal{E}, \mu)$ is a finite measure space and $\theta: E \rightarrow E$ is a measurable map. Prove that $\mu_{\theta}(A)=\mu\left(\theta^{-1}(A)\right)$ defines a measure on $(E, \mathcal{E})$.

  (b) Suppose that $\mathcal{A}$ is a $\pi$-system which generates $\mathcal{E}$. Using Dynkin's lemma, prove that $\theta$ is measure-preserving if and only if $\mu_{\theta}(A)=\mu(A)$ for all $A \in \mathcal{A}$.

  (c) State Birkhoff's ergodic theorem and the maximal ergodic lemma.

  (d) Consider the case $(E, \mathcal{E}, \mu)=([0,1), \mathcal{B}([0,1)), \mu)$ where $\mu$ is Lebesgue measure on $[0,1)$. Let $\theta:[0,1) \rightarrow[0,1)$ be the following map. If $x=\sum_{n=1}^{\infty} 2^{-n} \omega_{n}$ is the binary expansion of $x$ (where we disallow infinite sequences of $1 \mathrm{~s}$ ), then $\theta(x)=$ $\sum_{n=1}^{\infty} 2^{-n}\left(\omega_{n-1} \mathbf{1}_{n \in E}+\omega_{n+1} \mathbf{1}_{n \in O}\right)$ where $E$ and $O$ are respectively the even and odd elements of $\mathbb{N}$.

  (i) Prove that $\theta$ is measure-preserving. [You may assume that $\theta$ is measurable.]

  (ii) Prove or disprove: $\theta$ is ergodic.

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• Paper 1, Section II, J

  #Probability and Measure

  2017

  (a) Give the definition of the Borel $\sigma$-algebra on $\mathbb{R}$ and a Borel function $f: E \rightarrow \mathbb{R}$ where $(E, \mathcal{E})$ is a measurable space.

  (b) Suppose that $\left(f_{n}\right)$ is a sequence of Borel functions which converges pointwise to a function $f$. Prove that $f$ is a Borel function.

  (c) Let $R_{n}:[0,1) \rightarrow \mathbb{R}$ be the function which gives the $n$th binary digit of a number in $[0,1$ ) (where we do not allow for the possibility of an infinite sequence of 1 s). Prove that $R_{n}$ is a Borel function.

  (d) Let $f:[0,1)^{2} \rightarrow[0, \infty]$ be the function such that $f(x, y)$ for $x, y \in[0,1)^{2}$ is equal to the number of digits in the binary expansions of $x, y$ which disagree. Prove that $f$ is non-negative measurable.

  (e) Compute the Lebesgue measure of $f^{-1}([0, \infty))$, i.e. the set of pairs of numbers in $[0,1)$ whose binary expansions disagree in a finite number of digits.

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• Paper 2, Section II, G

  #Representation Theory

  2017

  In this question you may assume the following result. Let $\chi$ be a character of a finite group $G$ and let $g \in G$. If $\chi(g)$ is a rational number, then $\chi(g)$ is an integer.

  (a) If $a$ and $b$ are positive integers, we denote their highest common factor by $(a, b)$. Let $g$ be an element of order $n$ in the finite group $G$. Suppose that $g$ is conjugate to $g^{i}$ for all $i$ with $1 \leqslant i \leqslant n$ and $(i, n)=1$. Prove that $\chi(g)$ is an integer for all characters $\chi$ of $G$.

  [You may use the following result without proof. Let $\omega$ be an $n$th root of unity. Then

  is an integer.]

  Deduce that all the character values of symmetric groups are integers.

  (b) Let $G$ be a group of odd order.

  Let $\chi$ be an irreducible character of $G$ with $\chi=\bar{\chi}$. Prove that

  $$\left\langle\chi, 1_{G}\right\rangle=\frac{1}{|G|}(\chi(1)+2 \alpha),$$

  where $\alpha$ is an algebraic integer. Deduce that $\chi=1_{G}$.

  $$\begin{aligned} & \sum_{1 \leqslant i \leqslant n,} \omega^{i} \\ & \begin{aligned}&1 \leqslant i \leqslant n \\&(i, n)=1\end{aligned} \end{aligned}$$

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• Paper 3, Section II, G

  #Representation Theory

  2017

  (a) State Burnside's $p^{a} q^{b}$ theorem.

  (b) Let $P$ be a non-trivial group of prime power order. Show that if $H$ is a non-trivial normal subgroup of $P$, then $H \cap Z(P) \neq\{1\}$.

  Deduce that a non-abelian simple group cannot have an abelian subgroup of prime power index.

  (c) Let $\rho$ be a representation of the finite group $G$ over $\mathbb{C}$. Show that $\delta: g \mapsto \operatorname{det}(\rho(g))$ is a linear character of $G$. Assume that $\delta(g)=-1$ for some $g \in G$. Show that $G$ has a normal subgroup of index 2 .

  Now let $E$ be a group of order $2 k$, where $k$ is an odd integer. By considering the regular representation of $E$, or otherwise, show that $E$ has a normal subgroup of index $2 .$

  Deduce that if $H$ is a non-abelian simple group of order less than 80 , then $H$ has order 60 .

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• Paper 1, Section II, G

  #Representation Theory

  2017

  (a) Prove that if there exists a faithful irreducible complex representation of a finite group $G$, then the centre $Z(G)$ is cyclic.

  (b) Define the permutations $a, b, c \in S_{6}$ by

  $$a=\left(\begin{array}{lll} 1 & 2 & 3 \end{array}\right), b=\left(\begin{array}{lll} 4 & 5 & 6 \end{array}\right), c=\left(\begin{array}{ll} 2 & 3 \end{array}\right)(45),$$

  and let $E=\langle a, b, c\rangle$.

  (i) Using the relations $a^{3}=b^{3}=c^{2}=1, a b=b a, c^{-1} a c=a^{-1}$ and $c^{-1} b c=b^{-1}$, prove that $E$ has order 18 .

  (ii) Suppose that $\varepsilon$ and $\eta$ are complex cube roots of unity. Prove that there is a (matrix) representation $\rho$ of $E$ over $\mathbb{C}$ such that

  $$a \mapsto\left(\begin{array}{cc} \varepsilon & 0 \\ 0 & \varepsilon^{-1} \end{array}\right), \quad b \mapsto\left(\begin{array}{cc} \eta & 0 \\ 0 & \eta^{-1} \end{array}\right), \quad c \mapsto\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right)$$

  (iii) For which values of $\varepsilon, \eta$ is $\rho$ faithful? For which values of $\varepsilon, \eta$ is $\rho$ irreducible?

  (c) Note that $\langle a, b\rangle$ is a normal subgroup of $E$ which is isomorphic to $C_{3} \times C_{3}$. By inducing linear characters of this subgroup, or otherwise, obtain the character table of $E$.

  Deduce that $E$ has the property that $Z(E)$ is cyclic but $E$ has no faithful irreducible representation over $\mathbb{C}$.

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• Paper 4, Section II, G

  #Representation Theory

  2017

  Let $G=\mathrm{SU}(2)$ and let $V_{n}$ be the vector space of complex homogeneous polynomials of degree $n$ in two variables.

  (a) Prove that $V_{n}$ has the structure of an irreducible representation for $G$.

  (b) State and prove the Clebsch-Gordan theorem.

  (c) Quoting without proof any properties of symmetric and exterior powers which you need, decompose $\mathrm{S}^{2} V_{n}$ and $\Lambda^{2} V_{n}(n \geqslant 1)$ into irreducible $G$-spaces.

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• Paper 2, Section II, F

  #Riemann Surfaces

  2017

  Let $f$ be a non-constant elliptic function with respect to a lattice $\Lambda \subset \mathbb{C}$. Let $P$ be a fundamental parallelogram whose boundary contains no zeros or poles of $f$. Show that the number of zeros of $f$ in $P$ is the same as the number of poles of $f$ in $P$, both counted with multiplicities.

  Suppose additionally that $f$ is even. Show that there exists a rational function $Q(z)$ such that $f=Q(\wp)$, where $\wp$ is the Weierstrass $\wp$-function.

  Suppose $f$ is a non-constant elliptic function with respect to a lattice $\Lambda \subset \mathbb{C}$, and $F$ is a meromorphic antiderivative of $f$, so that $F^{\prime}=f$. Is it necessarily true that $F$ is an elliptic function? Justify your answer.

  [You may use standard properties of the Weierstrass $\wp$-function throughout.]

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• Paper 3, Section II, F

  #Riemann Surfaces

  2017

  Let $n \geqslant 2$ be a positive even integer. Consider the subspace $R$ of $\mathbb{C}^{2}$ given by the equation $w^{2}=z^{n}-1$, where $(z, w)$ are coordinates in $\mathbb{C}^{2}$, and let $\pi: R \rightarrow \mathbb{C}$ be the restriction of the projection map to the first factor. Show that $R$ has the structure of a Riemann surface in such a way that $\pi$ becomes an analytic map. If $\tau$ denotes projection onto the second factor, show that $\tau$ is also analytic. [You may assume that $R$ is connected.]

  Find the ramification points and the branch points of both $\pi$ and $\tau$. Compute the ramification indices at the ramification points.

  Assume that, by adding finitely many points, it is possible to compactify $R$ to a Riemann surface $\bar{R}$ such that $\pi$ extends to an analytic map $\bar{\pi}: \bar{R} \rightarrow \mathbb{C}_{\infty}$. Find the genus of $\bar{R}$ (as a function of $n$ ).

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• Paper 1, Section II, F

  #Riemann Surfaces

  2017

  By considering the singularity at $\infty$, show that any injective analytic map $f: \mathbb{C} \rightarrow \mathbb{C}$ has the form $f(z)=a z+b$ for some $a \in \mathbb{C}^{*}$ and $b \in \mathbb{C}$.

  State the Riemann-Hurwitz formula for a non-constant analytic map $f: R \rightarrow S$ of compact Riemann surfaces $R$ and $S$, explaining each term that appears.

  Suppose $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ is analytic of degree 2. Show that there exist Möbius transformations $S$ and $T$ such that

  $$S f T: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$$

  is the map given by $z \mapsto z^{2}$.

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• Paper 1, Section I, J

  #Statistical Modelling

  2017

  The dataset ChickWeights records the weight of a group of chickens fed four different diets at a range of time points. We perform the following regressions in $R$.

  

  (i) Which hypothesis test does the following command perform? State the degrees of freedom, and the conclusion of the test.

  

  (ii) Define a diagnostic plot that might suggest the logarithmic transformation of the response in fit2.

  (iii) Define the dashed line in the following plot, generated with the command plot(fit3). What does it tell us about the data point 579 ?

  

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• Paper 2, Section I, J

  #Statistical Modelling

  2017

  A statistician is interested in the power of a $t$-test with level $5 \%$ in linear regression; that is, the probability of rejecting the null hypothesis $\beta_{0}=0$ with this test under an alternative with $\beta_{0}>0$.

  (a) State the distribution of the least-squares estimator $\hat{\beta}_{0}$, and hence state the form of the $t$-test statistic used.

  (b) Prove that the power does not depend on the other coefficients $\beta_{j}$ for $j>0$.

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• Paper 3, Section I, J

  #Statistical Modelling

  2017

  For Fisher's method of Iteratively Reweighted Least-Squares and Newton-Raphson optimisation of the log-likelihood, the vector of parameters $\beta$ is updated using an iteration

  $$\beta^{(m+1)}=\beta^{(m)}+M\left(\beta^{(m)}\right)^{-1} U\left(\beta^{(m)}\right),$$

  for a specific function $M$. How is $M$ defined in each method?

  Prove that they are identical in a Generalised Linear Model with the canonical link function.

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• Paper 4, Section I, J

  #Statistical Modelling

  2017

  A Cambridge scientist is testing approaches to slow the spread of a species of moth in certain trees. Two groups of 30 trees were treated with different organic pesticides, and a third group of 30 trees was kept under control conditions. At the end of the summer the trees are classified according to the level of leaf damage, obtaining the following contingency table.

  

  Which of the following Generalised Linear Model fitting commands is appropriate for these data? Why? Describe the model being fit.

  

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• Paper 1, Section II, J

  #Statistical Modelling

  2017

  The Cambridge Lawn Tennis Club organises a tournament in which every match consists of 11 games, all of which are played. The player who wins 6 or more games is declared the winner.

  For players $a$ and $b$, let $n_{a b}$ be the total number of games they play against each other, and let $y_{a b}$ be the number of these games won by player $a$. Let $\tilde{n}_{a b}$ and $\tilde{y}_{a b}$ be the corresponding number of matches.

  A statistician analysed the tournament data using a Binomial Generalised Linear Model (GLM) with outcome $y_{a b}$. The probability $P_{a b}$ that $a$ wins a game against $b$ is modelled by

  $$\log \left(\frac{P_{a b}}{1-P_{a b}}\right)=\beta_{a}-\beta_{b},$$

  with an appropriate corner point constraint. You are asked to re-analyse the data, but the game-level results have been lost and you only know which player won each match.

  We define a new GLM for the outcomes $\tilde{y}_{a b}$ with $\tilde{P}_{a b}=\mathbb{E} \tilde{y}_{a b} / \tilde{n}_{a b}$ and $g\left(\tilde{P}_{a b}\right)=$ $\beta_{a}-\beta_{b}$, where the $\beta_{a}$ are defined in $(*)$. That is, $\beta_{a}-\beta_{b}$ is the log-odds that $a$ wins a game against $b$, not a match.

  Derive the form of the new link function $g$. [You may express your answer in terms of a cumulative distribution function.]

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• Paper 4, Section II, J

  #Statistical Modelling

  2017

  The dataset diesel records the number of diesel cars which go through a block of Hills Road in 6 disjoint periods of 30 minutes, between 8AM and 11AM. The measurements are repeated each day for 10 days. Answer the following questions based on the code below, which is shown with partial output.

  (a) Can we reject the model fit. 1 at a $1 \%$ level? Justify your answer.

  (b) What is the difference between the deviance of the models fit. 2 and fit.3?

  (c) Which of fit. 2 and fit. 3 would you use to perform variable selection by backward stepwise selection? Why?

  (d) How does the final plot differ from what you expect under the model in fit.2? Provide a possible explanation and suggest a better model.

  $>$ head (diesel)

  period num.cars day

  $\begin{array}{llll}1 & 1 & 69 & 1\end{array}$

  $\begin{array}{lllll}2 & 2 & 97 & 1\end{array}$

  $\begin{array}{llll}3 & 3 & 103 & 1\end{array}$

  $\begin{array}{llll}4 & 4 & 99 & 1\end{array}$

  $\begin{array}{llll}5 & 5 & 67 & 1\end{array}$

  $6 \quad 6 \quad 911$

  $>$ fit. $1=$ glm(num.cars period, data=diesel, family=poisson)

  summary (fit.1)

  Deviance Residuals:

  Min 1Q Median 3Q Max

  $\begin{array}{lllll}-4.0188 & -1.4837 & -0.2117 & 1.6257 & 4.5965\end{array}$

  Coefficients:

  Estimate Std. Error $z$ value $\operatorname{Pr}(>|z|)$

  (Intercept) $4.628535 \quad 0.029288158 .035<2 \mathrm{e}-16 * * *$

  period $-0.006073 \quad 0.007551-0.804 \quad 0.421$

  Signif. codes: 0 ? $* * *$ ? $0.001 ? * * ? 0.01$ ? $*$ ? $0.05$ ?.? $0.1$ ? ? 1

  (Dispersion parameter for poisson family taken to be 1)

  Null deviance: $262.36$ on 59 degrees of freedom

  Residual deviance: $261.72$ on 58 degrees of freedom

  AIC: $651.2$

  $>$ diesel$period.factor = factor(diesel$period)

  $>$ fit. $2=$ glm (num.cars period.factor, data=diesel, family=poisson)

  $\operatorname{summary}$ (fit.2)

  Coefficients:

  Estimate Std. Error z value $\operatorname{Pr}(>|z|)$

  Part II, $2017 \quad$ List of Questions

  [TURN OVER

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• Paper 4, Section II, D

  #Statistical Physics

  2017

  The van der Waals equation of state is

  $$p=\frac{k T}{v-b}-\frac{a}{v^{2}}$$

  where $p$ is the pressure, $v=V / N$ is the volume divided by the number of particles, $T$ is the temperature, $k$ is Boltzmann's constant and $a, b$ are positive constants.

  (i) Prove that the Gibbs free energy $G=E+p V-T S$ satisfies $G=\mu N$. Hence obtain an expression for $(\partial \mu / \partial p)_{T, N}$ and use it to explain the Maxwell construction for determining the pressure at which the gas and liquid phases can coexist at a given temperature.

  (ii) Explain what is meant by the critical point and determine the values $p_{c}, v_{c}, T_{c}$ corresponding to this point.

  (iii) By defining $\bar{p}=p / p_{c}, \bar{v}=v / v_{c}$ and $\bar{T}=T / T_{c}$, derive the law of corresponding states:

  $$\bar{p}=\frac{8 \bar{T}}{3 \bar{v}-1}-\frac{3}{\bar{v}^{2}} .$$

  (iv) To investigate the behaviour near the critical point, let $\bar{T}=1+t$ and $\bar{v}=1+\phi$, where $t$ and $\phi$ are small. Expand $\bar{p}$ to cubic order in $\phi$ and hence show that

  $$\left(\frac{\partial \bar{p}}{\partial \phi}\right)_{t}=-\frac{9}{2} \phi^{2}+\mathcal{O}\left(\phi^{3}\right)+t[-6+\mathcal{O}(\phi)] .$$

  At fixed small $t$, let $\phi_{l}(t)$ and $\phi_{g}(t)$ be the values of $\phi$ corresponding to the liquid and gas phases on the co-existence curve. By changing the integration variable from $p$ to $\phi$, use the Maxwell construction to show that $\phi_{l}(t)=-\phi_{g}(t)$. Deduce that, as the critical point is approached along the co-existence curve,

  $$\bar{v}_{\text {gas }}-\bar{v}_{\text {liquid }} \sim\left(T_{c}-T\right)^{1 / 2}$$

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• Paper 1, Section II, D

  #Statistical Physics

  2017

  Explain what is meant by the microcanonical ensemble for a quantum system. Sketch how to derive the probability distribution for the canonical ensemble from the microcanonical ensemble. Under what physical conditions should each type of ensemble be used?

  A paramagnetic solid contains atoms with magnetic moment $\boldsymbol{\mu}=\mu_{B} \mathbf{J}$, where $\mu_{B}$ is a positive constant and $\mathbf{J}$ is the intrinsic angular momentum of the atom. In an applied magnetic field $\mathbf{B}$, the energy of an atom is $-\boldsymbol{\mu} \cdot \mathbf{B}$. Consider $\mathbf{B}=(0,0, B)$. Each atom has total angular momentum $J \in \mathbb{Z}$, so the possible values of $J_{z}=m \in \mathbb{Z}$ are $-J \leqslant m \leqslant J$.

  Show that the partition function for a single atom is

  $$Z_{1}(T, B)=\frac{\sinh \left(x\left(J+\frac{1}{2}\right)\right)}{\sinh (x / 2)}$$

  where $x=\mu_{B} B / k T$.

  Compute the average magnetic moment $\left\langle\mu_{z}\right\rangle$ of the atom. Sketch $\left\langle\mu_{z}\right\rangle / J$ for $J=1$, $J=2$ and $J=3$ on the same graph.

  The total magnetization is $M_{z}=N\left\langle\mu_{z}\right\rangle$, where $N$ is the number of atoms. The magnetic susceptibility is defined by

  $$\chi=\left(\frac{\partial M_{z}}{\partial B}\right)_{T}$$

  Show that the solid obeys Curie's law at high temperatures. Compute the susceptibility at low temperatures and give a physical explanation for the result.

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• Paper 2, Section II, 34D

  #Statistical Physics

  2017

  (a) The entropy of a thermodynamic ensemble is defined by the formula

  $$S=-k \sum_{n} p(n) \log p(n)$$

  where $k$ is the Boltzmann constant. Explain what is meant by $p(n)$ in this formula. Write down an expression for $p(n)$ in the grand canonical ensemble, defining any variables you need. Hence show that the entropy $S$ is related to the grand canonical partition function $\mathcal{Z}(T, \mu, V)$ by

  $$S=k\left[\frac{\partial}{\partial T}(T \log \mathcal{Z})\right]_{\mu, V}$$

  (b) Consider a gas of non-interacting fermions with single-particle energy levels $\epsilon_{i}$.

  (i) Show that the grand canonical partition function $\mathcal{Z}$ is given by

  $$\log \mathcal{Z}=\sum_{i} \log \left(1+e^{-\left(\epsilon_{i}-\mu\right) /(k T)}\right)$$

  (ii) Assume that the energy levels are continuous with density of states $g(\epsilon)=A V \epsilon^{a}$, where $A$ and $a$ are positive constants. Prove that

  $$\log \mathcal{Z}=V T^{b} f(\mu / T)$$

  and give expressions for the constant $b$ and the function $f$.

  (iii) The gas is isolated and undergoes a reversible adiabatic change. By considering the ratio $S / N$, prove that $\mu / T$ remains constant. Deduce that $V T^{c}$ and $p V^{d}$ remain constant in this process, where $c$ and $d$ are constants whose values you should determine.

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• Paper 3, Section II, D

  #Statistical Physics

  2017

  (a) Describe the Carnot cycle using plots in the $(p, V)$-plane and the $(T, S)$-plane. In which steps of the cycle is heat absorbed or emitted by the gas? In which steps is work done on, or by, the gas?

  (b) An ideal monatomic gas undergoes a reversible cycle described by a triangle in the $(p, V)$-plane with vertices at the points $A, B, C$ with coordinates $\left(p_{0}, V_{0}\right),\left(2 p_{0}, V_{0}\right)$ and $\left(p_{0}, 2 V_{0}\right)$ respectively. The cycle is traversed in the order $A B C A$.

  (i) Write down the equation of state and an expression for the internal energy of the gas.

  (ii) Derive an expression relating $T d S$ to $d p$ and $d V$. Use your expression to calculate the heat supplied to, or emitted by, the gas along $A B$ and $C A$.

  (iii) Show that heat is supplied to the gas along part of the line $B C$, and is emitted by the gas along the other part of the line.

  (iv) Calculate the efficiency $\eta=W / Q$ where $W$ is the total work done by the cycle and $Q$ is the total heat supplied.

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• Paper 2, Section II, $27 \mathrm{~J}$

  #Stochastic Financial Models

  2017

  (a) What is a Brownian motion?

  (b) Let $\left(B_{t}, t \geqslant 0\right)$ be a Brownian motion. Show that the process $\tilde{B}_{t}:=\frac{1}{c} B_{c^{2} t}$, $c \in \mathbb{R} \backslash\{0\}$, is also a Brownian motion.

  (c) Let $Z:=\sup _{t \geqslant 0} B_{t}$. Show that $c Z \stackrel{(d)}{=} Z$ for all $c>0$ (i.e. $c Z$ and $Z$ have the same laws). Conclude that $Z \in\{0,+\infty\}$ a.s.

  (d) Show that $\mathbb{P}[Z=+\infty]=1$.

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• Paper 3, Section II, J

  #Stochastic Financial Models

  2017

  (a) State the fundamental theorem of asset pricing for a multi-period model.

  Consider a market model in which there is no arbitrage, the prices for all European put and call options are already known and there is a riskless asset $S^{0}=\left(S_{t}^{0}\right)_{t \in\{0, \ldots, T\}}$ with $S_{t}^{0}=(1+r)^{t}$ for some $r \geqslant 0$. The holder of a so-called 'chooser option' $C\left(K, t_{0}, T\right)$ has the right to choose at a preassigned time $t_{0} \in\{0,1, \ldots, T\}$ between a European call and a European put option on the same asset $S^{1}$, both with the same strike price $K$ and the same maturity $T$. [We assume that at time $t_{0}$ the holder will take the option having the higher price at that time.]

  (b) Show that the payoff function of the chooser option is given by

  $$C\left(K, t_{0}, T\right)= \begin{cases}\left(S_{T}^{1}-K\right)^{+} & \text {if } S_{t_{0}}^{1}>K(1+r)^{t_{0}-T} \\ \left(K-S_{T}^{1}\right)^{+} & \text {otherwise }\end{cases}$$

  (c) Show that the price $\pi\left(C\left(K, t_{0}, T\right)\right)$ of the chooser option $C\left(K, t_{0}, T\right)$ is given by

  $$\pi\left(C\left(K, t_{0}, T\right)\right)=\pi(E C(K, T))+\pi\left(E P\left(K(1+r)^{t_{0}-T}, t_{0}\right)\right),$$

  where $\pi(E C(K, T))$ and $\pi(E P(K, T))$ denote the price of a European call and put option, respectively, with strike $K$ and maturity $T$.

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• Paper 4, Section II, J

  #Stochastic Financial Models

  2017

  (a) Describe the (Cox-Ross-Rubinstein) binomial model. When is the model arbitragefree? How is the equivalent martingale measure characterised in this case?

  (b) What is the price and the hedging strategy for any given contingent claim $C$ in the binomial model?

  (c) For any fixed $0<t<T$ and $K>0$, the payoff function of a forward-start-option is given by

  $$\left(\frac{S_{T}^{1}}{S_{t}^{1}}-K\right)^{+}$$

  Find a formula for the price of the forward-start-option in the binomial model.

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• Paper 1, Section II, J

  #Stochastic Financial Models

  2017

  (a) What does it mean to say that $\left(X_{n}, \mathcal{F}_{n}\right)_{n \geqslant 0}$ is a martingale?

  (b) Let $\Delta_{0}, \Delta_{1}, \ldots$ be independent random variables on $(\Omega, \mathcal{F}, \mathbb{P})$ with $\mathbb{E}\left[\left|\Delta_{i}\right|\right]<\infty$ and $\mathbb{E}\left[\Delta_{i}\right]=0, i \geqslant 0$. Further, let

  $$X_{0}=\Delta_{0} \quad \text { and } \quad X_{n+1}=X_{n}+\Delta_{n+1} f_{n}\left(X_{0}, \ldots, X_{n}\right), \quad n \geqslant 0$$

  where

  $$f_{n}\left(x_{0}, \ldots, x_{n}\right)=\frac{1}{n+1} \sum_{i=0}^{n} x_{i}$$

  Show that $\left(X_{n}\right)_{n \geqslant 0}$ is a martingale with respect to the natural filtration $\mathcal{F}_{n}=$ $\sigma\left(X_{0}, \ldots, X_{n}\right)$.

  (c) State and prove the optional stopping theorem for a bounded stopping time $\tau$.

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• Paper 2, Section I, F

  #Topics In Analysis

  2017

  Are the following statements true or false? Give reasons, quoting any theorems that you need.

  (i) There is a sequence of polynomials $P_{n}$ with $P_{n}(t) \rightarrow \sin t$ uniformly on $\mathbb{R}$ as $n \rightarrow \infty$.

  (ii) If $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous, then there is a sequence of polynomials $Q_{n}$ with $Q_{n}(t) \rightarrow f(t)$ for each $t \in \mathbb{R}$ as $n \rightarrow \infty$.

  (iii) If $g:[1, \infty) \rightarrow \mathbb{R}$ is continuous with $g(t) \rightarrow 0$ as $t \rightarrow \infty$, then there is a sequence of polynomials $R_{n}$ with $R_{n}(1 / t) \rightarrow g(t)$ uniformly on $[1, \infty)$ as $n \rightarrow \infty$.

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• Paper 4, Section I, F

  #Topics In Analysis

  2017

  If $x \in(0,1]$, set

  $$x=\frac{1}{N(x)+T(x)}$$

  where $N(x)$ is an integer and $1>T(x) \geqslant 0$. Let $N(0)=T(0)=0$.

  If $x$ is also irrational, write down the continued fraction expansion in terms of $N T^{j}(x)\left(\right.$ where $\left.N T^{0}(x)=N(x)\right)$.

  Let $X$ be a random variable taking values in $[0,1]$ with probability density function

  $$f(x)=\frac{1}{(\log 2)(1+x)}$$

  Show that $T(X)$ has the same distribution as $X$.

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• Paper 1, Section I, $2 F$

  #Topics In Analysis

  2017

  State Liouville's theorem on the approximation of algebraic numbers by rationals.

  Suppose that we have a sequence $\zeta_{n}$ with $\zeta_{n} \in\{0,1\}$. State and prove a necessary and sufficient condition on the $\zeta_{n}$ for

  $$\sum_{n=0}^{\infty} \zeta_{n} 10^{-n !}$$

  to be transcendental.

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• Paper 3, Section I, $2 F$

  #Topics In Analysis

  2017

  (a) Suppose that $g: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ is a continuous function such that there exists a $K>0$ with $\|g(\mathbf{x})-\mathbf{x}\| \leqslant K$ for all $\mathbf{x} \in \mathbb{R}^{2}$. By constructing a suitable map $f$ from the closed unit disc into itself, show that there exists a $\mathbf{t} \in \mathbb{R}^{2}$ with $g(\mathbf{t})=\mathbf{0}$.

  (b) Show that $g$ is surjective.

  (c) Show that the result of part (b) may be false if we drop the condition that $g$ is continuous.

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• Paper 2, Section II, F

  #Topics In Analysis

  2017

  State and prove Baire's category theorem for complete metric spaces. Give an example to show that it may fail if the metric space is not complete.

  Let $f_{n}:[0,1] \rightarrow \mathbb{R}$ be a sequence of continuous functions such that $f_{n}(x)$ converges for all $x \in[0,1]$. Show that if $\epsilon>0$ is fixed we can find an $N \geqslant 0$ and a non-empty open interval $J \subseteq[0,1]$ such that $\left|f_{n}(x)-f_{m}(x)\right| \leqslant \epsilon$ for all $x \in J$ and all $n, m \geqslant N$.

  Let $g:[0,1] \rightarrow \mathbb{R}$ be defined by

  $$g(x)= \begin{cases}1 & \text { if } x \text { is rational } \\ 0 & \text { if } x \text { is irrational. }\end{cases}$$

  Show that we cannot find continuous functions $g_{n}:[0,1] \rightarrow \mathbb{R}$ with $g_{n}(x) \rightarrow g(x)$ for each $x \in[0,1]$ as $n \rightarrow \infty .$

  Define a sequence of continuous functions $h_{n}:[0,1] \rightarrow \mathbb{R}$ and a discontinuous function $h:[0,1] \rightarrow \mathbb{R}$ with $h_{n}(x) \rightarrow h(x)$ for each $x \in[0,1]$ as $n \rightarrow \infty$.

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• Paper 4, Section II, $11 F$

  #Topics In Analysis

  2017

  (a) Suppose that $\gamma:[0,1] \rightarrow \mathbb{C}$ is continuous with $\gamma(0)=\gamma(1)$ and $\gamma(t) \neq 0$ for all $t \in[0,1]$. Show that if $\gamma(0)=|\gamma(0)| \exp \left(i \theta_{0}\right)$ (with $\theta_{0}$ real) we can define a continuous function $\theta:[0,1] \rightarrow \mathbb{R}$ such that $\theta(0)=\theta_{0}$ and $\gamma(t)=|\gamma(t)| \exp (i \theta(t))$. Hence define the winding number $w(\gamma)=w(0, \gamma)$ of $\gamma$ around 0 .

  (b) Show that $w(\gamma)$ can take any integer value.

  (c) If $\gamma_{1}$ and $\gamma_{2}$ satisfy the requirements of the definition, and $\left(\gamma_{1} \times \gamma_{2}\right)(t)=\gamma_{1}(t) \gamma_{2}(t)$, show that

  $$w\left(\gamma_{1} \times \gamma_{2}\right)=w\left(\gamma_{1}\right)+w\left(\gamma_{2}\right)$$

  (d) If $\gamma_{1}$ and $\gamma_{2}$ satisfy the requirements of the definition and $\left|\gamma_{1}(t)-\gamma_{2}(t)\right|<\left|\gamma_{1}(t)\right|$ for all $t \in[0,1]$, show that

  $$w\left(\gamma_{1}\right)=w\left(\gamma_{2}\right)$$

  (e) State and prove a theorem that says that winding number is unchanged under an appropriate homotopy.

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• Paper 2, Section II, B

  #Waves

  2017

  Show that, for a one-dimensional flow of a perfect gas (with $\gamma>1$ ) at constant entropy, the Riemann invariants $R_{\pm}=u \pm 2\left(c-c_{0}\right) /(\gamma-1)$ are constant along characteristics $d x / d t=u \pm c .$

  Define a simple wave. Show that in a right-propagating simple wave

  $$\frac{\partial u}{\partial t}+\left(c_{0}+\frac{1}{2}(\gamma+1) u\right) \frac{\partial u}{\partial x}=0$$

  In some circumstances, dissipative effects may be modelled by

  $$\frac{\partial u}{\partial t}+\left(c_{0}+\frac{1}{2}(\gamma+1) u\right) \frac{\partial u}{\partial x}=-\alpha u$$

  where $\alpha$ is a positive constant. Suppose also that $u$ is prescribed at $t=0$ for all $x$, say $u(x, 0)=u_{0}(x)$. Demonstrate that, unless a shock develops, a solution of the form

  $$u(x, t)=u_{0}(\xi) e^{-\alpha t}$$

  can be found, where, for each $x$ and $t, \xi$ is determined implicitly as the solution of the equation

  $$x-c_{0} t=\xi+\frac{\gamma+1}{2 \alpha}\left(1-e^{-\alpha t}\right) u_{0}(\xi)$$

  Deduce that, despite the presence of dissipative effects, a shock will still form at some $(x, t)$ unless $\alpha>\alpha_{c}$, where

  $$\alpha_{c}=\frac{1}{2}(\gamma+1) \max _{u_{0}^{\prime}<0}\left|u_{0}^{\prime}(\xi)\right|$$

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• Paper 1, Section II, B

  #Waves

  2017

  Derive the wave equation governing the pressure disturbance $\tilde{p}$, for linearised, constant entropy sound waves in a compressible inviscid fluid of density $\rho_{0}$ and sound speed $c_{0}$, which is otherwise at rest.

  Consider a harmonic acoustic plane wave with wavevector $\mathbf{k}_{I}=k_{I}(\sin \theta, \cos \theta, 0)$ and unit-amplitude pressure disturbance. Determine the resulting velocity field $\mathbf{u}$.

  Consider such an acoustic wave incident from $y<0$ on a thin elastic plate at $y=0$. The regions $y<0$ and $y>0$ are occupied by gases with densities $\rho_{1}$ and $\rho_{2}$, respectively, and sound speeds $c_{1}$ and $c_{2}$, respectively. The kinematic boundary conditions at the plate are those appropriate for an inviscid fluid, and the (linearised) dynamic boundary condition is

  $$m \frac{\partial^{2} \eta}{\partial t^{2}}+B \frac{\partial^{4} \eta}{\partial x^{4}}+[\tilde{p}(x, 0, t)]_{-}^{+}=0$$

  where $m$ and $B$ are the mass and bending moment per unit area of the plate, and $y=\eta(x, t)$ (with $\left|\mathbf{k}_{I} \eta\right| \ll 1$ ) is its perturbed position. Find the amplitudes of the reflected and transmitted pressure perturbations, expressing your answers in terms of the dimensionless parameter

  $$\beta=\frac{k_{I} \cos \theta\left(m c_{1}^{2}-B k_{I}^{2} \sin ^{4} \theta\right)}{\rho_{1} c_{1}^{2}}$$

  (i) If $\rho_{1}=\rho_{2}=\rho_{0}$ and $c_{1}=c_{2}=c_{0}$, under what condition is the incident wave perfectly transmitted?

  (ii) If $\rho_{1} c_{1} \gg \rho_{2} c_{2}$, comment on the reflection coefficient, and show that waves incident at a sufficiently large angle are reflected as if from a pressure-release surface (i.e. an interface where $\tilde{p}=0$ ), no matter how large the plate mass and bending moment may be.

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• Paper 3, Section II, B

  #Waves

  2017

  Waves propagating in a slowly-varying medium satisfy the local dispersion relation $\omega=\Omega(\mathbf{k} ; \mathbf{x}, t)$ in the standard notation. Derive the ray-tracing equations

  $$\frac{d x_{i}}{d t}=\frac{\partial \Omega}{\partial k_{i}}, \quad \frac{d k_{i}}{d t}=-\frac{\partial \Omega}{\partial x_{i}}, \quad \frac{d \omega}{d t}=\frac{\partial \Omega}{\partial t}$$

  governing the evolution of a wave packet specified by $\varphi(\mathbf{x}, t)=A(\mathbf{x}, t ; \varepsilon) e^{i \theta(\mathbf{x}, t) / \varepsilon}$, where $0<\varepsilon \ll 1$. A formal justification is not required, but the meaning of the $d / d t$ notation should be carefully explained.

  The dispersion relation for two-dimensional, small amplitude, internal waves of wavenumber $\mathbf{k}=(k, 0, m)$, relative to Cartesian coordinates $(x, y, z)$ with $z$ vertical, propagating in an inviscid, incompressible, stratified fluid that would otherwise be at rest, is given by

  $$\omega^{2}=\frac{N^{2} k^{2}}{k^{2}+m^{2}},$$

  where $N$ is the Brunt-Väisälä frequency and where you may assume that $k>0$ and $\omega>0$. Derive the modified dispersion relation if the fluid is not at rest, and instead has a slowly-varying mean flow $(U(z), 0,0)$.

  In the case that $U^{\prime}(z)>0, U(0)=0$ and $N$ is constant, show that a disturbance with wavenumber $\mathbf{k}=(k, 0,0)$ generated at $z=0$ will propagate upwards but cannot go higher than a critical level $z=z_{c}$, where $U\left(z_{c}\right)$ is equal to the apparent wave speed in the $x$-direction. Find expressions for the vertical wave number $m$ as $z \rightarrow z_{c}$ from below, and show that it takes an infinite time for the wave to reach the critical level.

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• Paper 4, Section II, 38B

  #Waves

  2017

  Consider the Rossby-wave equation

  $$\frac{\partial}{\partial t}\left(\frac{\partial^{2}}{\partial x^{2}}-\ell^{2}\right) \varphi+\beta \frac{\partial \varphi}{\partial x}=0,$$

  where $\ell>0$ and $\beta>0$ are real constants. Find and sketch the dispersion relation for waves with wavenumber $k$ and frequency $\omega(k)$. Find and sketch the phase velocity $c(k)$ and the group velocity $c_{g}(k)$, and identify in which direction(s) the wave crests travel, and the corresponding direction(s) of the group velocity.

  Write down the solution with initial value

  $$\varphi(x, 0)=\int_{-\infty}^{\infty} A(k) e^{i k x} d k$$

  where $A(k)$ is real and $A(-k)=A(k)$. Use the method of stationary phase to obtain leading-order approximations to $\varphi(x, t)$ for large $t$, with $x / t$ having the constant value $V$, for

  (i) $0<V<\beta / 8 \ell^{2}$,

  (ii) $-\beta / \ell^{2}<V \leqslant 0$,

  where the solutions for the stationary points should be left in implicit form. [It is helpful to note that $\omega(-k)=-\omega(k)$.]

  Briefly discuss the nature of the solution for $V>\beta / 8 \ell^{2}$ and $V<-\beta / \ell^{2}$. [Detailed calculations are not required.]

  [Hint: You may assume that

  $$\int_{-\infty}^{\infty} e^{\pm i \gamma u^{2}} d u=\left(\frac{\pi}{\gamma}\right)^{\frac{1}{2}} e^{\pm i \pi / 4}$$

  for $\gamma>0 .]$

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